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Mental  Arithmetic 


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IN  MEMORIAM 
FLOR1AN  CAJOR1 


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PEEFACE. 


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In  the  solution  of  problems  there  are  two  distinct  steps  —  the  selection 
of  the  operations,  and  their  performance.  Mental  and  written  arith- 
metic agree  in  that  the  choice  of  operations  is  determined  in  the  same 
manner  ;  they  differ  in  that  the  operations  are  wholly  mental  in  the  one, 
while  external  aids  are  used  in  the  other.  Mental  arithmetic  should, 
therefore,  embrace  all  cases  in  written  arithmetic  except  those  which 
teach  how  to  add,  subtract,  multiply,  and  divide  large  numbers.  This 
arithmetic  is  intended  as  a  drill-book  in  which  the  principles  of  written 
arithmetic,  except  those  mentioned  above,  shall  be  concisely  stated  and 
illustrated.  The  examples  and  problems  are  such  as  the  average  mind 
should  be  able  to  solve  readily  without  a  pencil. 

He  who  teaches  by  the  printed  page  must  use  every  artifice  of  arrange- 
ment to  make  his  statements  clear  and  attractive.  The  placing  of  prin- 
ciples and  illustrations  in  parallel  columns  aids  the  student  to  grasp  the 
subject  as  a  whole,  since  each  column  may  be  read  independently,  and 
each  conveys  the  same  thought  in  a  different  manner.  The  beginning 
of  each  subject  at  the  top  of  a  page,  the  systematic  placing  of  explana- 
tions and  directions  under  exercises,  and  the  continuous  numbering  of  all 
the  examples  in  a  chapter,  aid  the  teacher  to  announce  and  the  pupil  to 
understand  the  requirements. 

In  Addition,  the  combination  method  is  made  prominent.  The  number 
of  seconds  which  should  be  required  for  the  solution  of  each  example  is 
stated  after  each  exercise.  Since  ninety  per  cent  of  all  arithmetical 
computation  in  the  work-shop,  farm,  and  counting-room  is  Addition,  this 
subject  cannot  be  too  zealously  pressed.  Many  who  have  broken  the 
habit,  in  adding,  of  saying  "  6  and  8  are  14  and  6  are  20,"  are  still  saying 
in  subtracting,  "  6  from  10  leaves  4  " ;  in  multiplying,  "  9  times  8  are  72, 
and  4  are  76  "  ;  and  in  dividing,  "  12  +  5  =  2  and  2  remaining."     Special 

3 


4  PREFACE. 

stress  is  laid  upon  the  importance,  in  performing  operations,  of  dropping 
all  unnecessary  words,  since  the  mind  reaches  results  much  more  rapidly 
without  them. 

In  factoring,  the  introduction  of  a  new  conception,  that  of  numbers 
severally  prime  to  each  other,  will  be  appreciated  by  experts,  and  cannot 
fail  to  benefit  learners,  because  it  obviates  the  cumbersome  expression  of 
numbers  by  their  prime  factors.  Those  who,  in  dividing  fractions,  have 
never  practiced  mentally  the  method  largely  used  in  Europe,  will  be 
delighted  with  the  ease  by  which  results  can  be  obtained. 

Attention  is  called  to  the  presentation  of  the  Metric  System.  By 
memorizing  the  table  of  submultiples  and  the  table  of  units,  the  student 
acquires  the  principles  of  the  whole  subject,  and  will  only  need  practice 
to  master  it. 

Percentage  is  taught  without  rules  or  formulae,  and  without  the  use  of 
the  terms  base,  amount,  and  difference,  although  one  page  is  devoted  to  them 
after  the  subject  has  been  completed.  The  student  comes  to  see  clearly 
that  the  various  exercises  in  percentage  do  not  need  special  rules,  but  are 
familiar  cases  slightly  modified  since  the  symbol  "  %  "  is  used  instead  of 
hundredths.  Interest  is  taught  by  the  6  %  method  and  by  the  modification 
of  this  method  in  general  use  among  bankers. 

The  practical  exercises  "  at  the  lumber  yard,"  "  at  the  carpet  store," 
etc.,  are  to  drill  the  student  in  methods  daily  used  at  such  places.  Men- 
suration has  been  developed  with  a  view  of  showing  the  necessity  for  the 
existence  of  the  various  forms,  their  relations,  and  their  limitations. 

Few  principles  are  presented,  but  these  few  are  the  keys  to  all  depart- 
ments of  the  science.  Let  it  be  remembered,  that  he  who  relies  upon 
thousands  of  special  rules  is  but  a  pygmy  beside  the  giant  who  can  apply 
a  score  of  general  principles  to  millions  of  particulars. 

M.  A.  BAILEY. 

State  Normal  School  of  Kansas. 


15  IS"7 


TABLE  OF  CONTENTS. 


PAGE 

Addition 7 

Combinations  —  Two  Figures 8 

Combinations  —  Three  Figures 14 

Combinations  —  Four  Figures 15 

Addends 16 

Problems 18 

Subtraction 21 

Problems .        .  26 

Multiplication 29 

Problems 33 

Division 36 

Precedence  of  Signs 41 

Parenthesis  or  Bar 42 

Problems -.        .45 

Factoring 48 

Multiplication  and  Division 52 

Greatest  Common  Divisor 54 

Least  Common  Multiple      .        .        .   * 55 

Common  Fractions 56 

First  Conception  —  An  Expression  of  Division         ....  56 

Second  Conception  —  One  or  More  of  the  Equal  Parts  of  a  Unit      .  57 

Change  of  Form  —  To  Higher  Terms 58 

Change  of  Form  —  To  Lower  Terms  . 59 

Change  of  Form  —  To  a  Whole  or  Mixed  Number    .        .  .60 

Addition  and  Subtraction 61 

Multiplication  —  Universal  Case 62 

Division  —  Universal  Case 63 

Problems 64 

Decimals 72 

Reduction  —  Common  Fractions  to  Decimals 74 

Reduction  —  Decimals  to  Common  Fractions 75 

Per  Cent 76 

Short  Methods 77 

Denominate  Numbers  —  English  Tables 81 

Money,  Weights 81 

Long  Measure 82 

5 


CONTENTS. 


PAGE 

Denominate  Numbers  —  English  Tables  (Continued). 

Square  and  Cubic  Measures 83 

Capacity,  Time 84 

Circular  Measure,  Counting,  Paper,  Equivalents       ....  86 

Exercises  in  Tables 87 

Reduction  in  Same  Table 90 

Reduction  Table  to  Table   .        . 92 

Denominate  Numbers — Metric  System       .         .         .         .         .         .94 

Practical  Questions     . 98 

Reduction  —  Metric 99 

Reduction  —  English  and  Metric 100 

Percentage 101 

Reduction 101 

The  Operation  Directly  Stated 102 

Operations  to  be  Determined 104 

Profit  and  Loss    .•                . 109 

Commission 112 

Interest 115 

Simple  Interest 116 

Trade  Discount 123 

True  Discount 124 

Bank  Discount 125 

Stocks 126 

Practical  Exercises 131 

At  the  Lumber  Yard 131 

Measurement  of  Logs 135 

At  the  Carpet  Store 136 

With  the  Paper  Hanger 137 

Average 138 

Involution  and  Evolution 139 

Proportion 140 

Mensuration      . 141 

One  Dimension 141 

Two  Dimensions 142 

Three  Dimensions 147 

Similarity ' 150 

Miscellaneous 152 

Arithmetical  Progression    . .        .  152 

Geometrical  Progression 153 

Specific  Gravity 154 

Zero  and  Infinity 155 

General  Review  Exercises 156 


AMERICAN  MENTAL  ARITHMETIC. 


ADDITION. 


Addition  is  indicated  by  the 
sign  +  . 

The  numbers  to  be  united  are 
addends;  the  result,  the  sum  or 
amount. 

The  sign  of  equality  is  = . 

The  sum  of  two  or  more  num- 
bers may  be  found  by  counting. 

Addition  is  a  process  shorter 
than  counting  for  finding  the 
sum  of  numbers. 

A  number  may  be  written  by 
the  decimal  notation  or  by  its 
addends. 

A  number  may  be  spelled  by 
naming  its  addends,  just  as  a  word 
is  spelled  by  naming  its  letters. 

A  number  may  be  spelled  in 
several  different  ways. 


Illustration. 

6  +  4  =  10 

read 

6  plus  4  equals  10. 

6  and  4,  addends. 

10,  sum  or  amount. 

To  find  the  sum  of  6  and 
4  by  counting. 

Counting  to  6  and  mak- 
ing a  mark  at  each  count, 
llllll;  counting  to  4  and 
making  a  mark  at  each 
count,  llllll  III/;  count- 
ing the  result,  we  have  10. 

Ten  may  be  written 

in  5    6     7     8    9 

10;   or  5,   4,    a,    2,    r 

Ten,  as  written  above,  is 
spelled  five  five,  six  four,  seven 
three,  eight  two,  or  nine  one. 


AMERICAN  MENTAL   ARITHMETIC. 


§  1.   Combinations  —  Two  Figures. 

There  are  45  different  combinations  of  figures  taking  two 
at  a  time,  viz. : 

999999999888888887777  77 
9'  8'  7'  6'  5'  4'  3'  2'  1'  8'  7'  6'  5'  4'  3'  2'  1'  f   6'  5'  4'  3'  2' 
76666665555544443  33221 
1'  6'  5'  4'  3'  2'  1'  5'  4'  3'  2'  1'  4'  3'  2'  1'  3'  2'  1'  2'  1'  1' 

These  combinations  are  shown  in  the  following  table,  and 
should  be  thoroughly  memorized. 


The 
Number. 

Combinations. 

The 
Number. 

Combinations. 

The 
Number. 

Combinations. 

2 

1 
1 

8 

4     5     6     7 
4'    3'   2'    1 

14 

7     8     9 
T   6'    5 

3 

2 
1 

9 

5    6     7     8 
4'    3'   2'    1 

15 

8    9 

7'   6 

4 

2     3 

2'    1 

10 

5    6     7     8    9 
5'   4'    3'   2'    1 

16 

8    9 

8'    7 

5 

3    4 

2'    1 

11 

6     7    8    9 
5'   4'   3'   2 

17 

9 

8 

6 

3    4    5 

3'    2'    1 

12 

6     7     8    9 
6'    5'    4'    3 

18 

9 
9 

7 

4    5    6 
3'    2'    1 

13 

7     8     9 
6'    5'    4 

1  2  2   3 

Memorize  thus :  2,  -.  (one  one) ;  3,  ..  (two  one) ;  4,  „,  n  (iwo  £w?o,  or,  ^ree 

u»e) ;  etc. 

1.  What  are  the  combinations  whose  sum  is  10  ? 

2.  What  are  the  combinations  whose  sum  is  12?  18?  9?  4? 

3.  What  are  the  combinations  whose  sum  is  17?  5?  11?  3? 

4.  What  are  the  combinations  whose  sum  is  2?  16?  6?  15? 

5.  What  are  the  combinations  whose  sum  is  7  ?  14?  8?  13  r' 

State  the  answers  without  reading  the  questions  aloud. 
5    6     7     8    9 


Ex.  1. 


5'   4' 


2'  r 


ADDITION  9 

Name  the  combinations  whose  sum  is  i 

6.  Two,  three,  four,  five,  six,  seven,  eight,  nine,  ten, 
eleven,  twelve,  thirteen,  fourteen,  fifteen,  sixteen,  seventeen, 
eighteen. 

7.  Eighteen,  seventeen,  sixteen,  fifteen,  fourteen,  thirteen, 
twelve,  eleven,  ten,  nine,  eight,  seven,  six,  five,  four,  three,  two. 

8.  Ten,  twelve,  fifteen,  two,  six,  eleven,  nine,  sixteen, 
three,  thirteen,  four,  seventeen,  five,  eighteen,  seven,  fourteen, 
eight. 

State  the  answers  without  reading  the  questions  aloud. 

i?      .7    9     9     8    9     8     9       . 
■bx-  7>  0;   8;  8'    7;   7'   6'  etc' 

Read  the  sums : 

Q97847678632467 
y-  9'  6'  5'  3'  2'  5'  8'  8'  4'  f  8'  4'  9'  6' 

_n  56972484458377 
10"  5'  3'  2'  2'  6'  4'  8'  1'  3'  8'  T    3'  4'  5* 

..44389612794624 
**:  8'  6'  5'  9'  9'  8'  1'  1'  7'  5'  9'  7'  2'  2* 

Ex.  9.  18,  13,  13,  7,  9,  etc.     Do  not  say  9  and  9  are  18. 


Read  the  sums : 

12.  9  +  9,  3  +  2,  6-}-6,  2  +  2,  5  +  5,  4  +  3,  7  +  6,  9  +  8,  6+5, 
2  +  1,  8  +  8,  7  +  7,  1  +  1,  5  +  4,  8  +  7,  4  +  4,  3  +  3. 

13.  9  +  7,  6  +  2,  9  +  4,  8  +  6,  9  +  6,  7  +  2,  6  +  4,  5  +  1,  9  +  5, 
7  +  5,8  +  4,5  +  3,8  +  5,  5  +  2,  7  +  1,  8  +  3,  7  +  3,  8  +  2,  6  +  3, 

6  +  1,4  +  2,9  +  3,7  +  4,4  +  1. 

14.  8  +  9,7  +  6,4  +  5,2  +  3,7  +  8,  8  +  4,  7  +  9,  7  +  7,  8  +  4, 

7  +  2,  8  +  7,  9  +  9,  8  +  8. 

Do  not  say  9  plus  9  equals  18,  but  state  sums  directly. 
Ex.  12.   18,  5,  12,  etc. 


10  AMERICAN  MENTAL   ARITHMETIC. 

To  1,  3,  6,  8,  9,  4,  7,  5,  2 

15.  Add  9.  18. 

16.  Add  8.  19. 

17.  Add  7.  20. 
Ex.  15.    10,  12,  15,  17,  18,  etc. 

State  the  results : 


Add  6. 

21.  Add  3. 

Add  5. 

22.   Add  2, 

Add  4. 

23.   Add  1, 

24. 

8  +  9  = 

25. 

7  +  8  = 

26. 

8  +  4  = 

27. 

Q  +  5  = 

28. 

5  +  9= 

29. 

3  +  7  = 

30. 

6  +  3  = 

31. 

4  +  6  = 

32.    6  +  9  = 

40.   9  +  5  = 

33.    4  +  7  = 

41.   3  +  9  = 

34.    6  +  8  = 

42.    9  +  2  = 

35.    9  +  9  = 

43.    6  +  6  = 

36.    6  +  7  = 

44.    8  +  8  = 

37.    7  +  5  = 

45.    4  +  9  = 

38.    8  +  5  = 

46.    3  +  8  = 

39.    7  +  9  = 

47.     7  +  7  = 

nd7?     6and5? 

4  and  9?    3  and  7? 

48.  How  many  are  9  and  7 

5  and  8?     7  and  2?     8  and  6? 

49.  How  many  are  7  and  6 ?    8and3?    9and5?    8and8? 

6  and  2?     4  and  5?     5  and  3?     7  and  9?     8  and  8?     4  and 
9?     4  and  3? 

50.  How  many  are  9  and  9?  6  and  4  ?  5  and  7  ?  8  and  9  ? 
3  and  6  ?  5  and  2  ?  9  and  6  ?  8  and  7  ?  7  and  7  ?  6  and 
6?     5  and  5? 

51.  Read  as  rapidly  as  possible  12,  16,  14,  13,  11,  18,  14, 
16,  17. 

52.  Read  as  rapidly  as  possible 

698759989 
6'    7'    6'    6'    6'    9'    5'    8'    8' 

The  student  has  mastered  these  combinations  when  he  can  read  numbers 
as  expressed  in  Ex.  52  as  rapidly  as  he  can  read  numbers  as  expressed  in 
Ex.  51. 


ADDITION.  11 

§  2.    In  General. 

Declare  the  sums: 

—     8   28   38   58   78   88  _     8   48   68   98   78   58 

53'    9'    9'   9'   9'    9'    9*  56'    6'   6'    6'    6'   6'    6* 


9   89   39   49   59   79 
4'   4'   4'   4'   4'   4* 


8   48   88   78   58   28 
8'    8'   8'   8'    8'   8' 


54.    _,    _ ,    „y    -)    -  >    „  •  57. 

RR     9   89   29   79   39   69  _Q 

55'    9'    9'    9'   9'    9'    9*  58' 

Ex.  53.    17,  37,  47,  etc. 

/What  is  the  right  hand  figure  of  the  sum, 

59.  When  9  is  added  to  a  number  ending  in  9?  8?  3?  7? 
4?  2?  5?  6?  1? 

60.  When  8  is  added  to  a  number  ending  in  8?  2?  6?  3? 
5?  9?  1?  4?  7? 

61.  When  7  is  added  to  a  number  ending  in  1?  3?  5?  7? 
9?  2?  4?  6?  8? 

62.  When  6  is  added  to  a  number  ending  in  6?  3?  8?  2? 
4?  1?  5?  7?  9?     When  5  is  added? 

63.  When  4  is  added  to  a  number  ending  in  2?  6?  4?  8? 
1?  9?  5?  7?  3?     When  3  is  added? 

64.  When  2  is  added  to  a  number  ending  in  3?  5?  1?  7? 
4?  9?  2?  6?  8?     When  1  is  added? 

Ex.  59.    8,  7,  2,  6,  3,  1,  4,  5,  0. 

Beginning  with  1,  count  as  rapidly  as  possible  to  about  100  : 

65.  By  9.  68.  By  6.  71.  By  3. 

66.  By  8.  69.  By  5.  72.  By  2. 

67.  By  7.  70.  By  4.  73.  By  1. 

Ex.  65.    1,  10,  19,  28,  37,  etc. 


12  AMERICAN   MENTAL   ARITHMETIC. 

Add: 

74.  3,  7,  6,  8,  9,  2,  7,  8,  5,  1,  2,  3,  4,  5,  6,  7,  8„9, 5,  9,  3. 

75.  4,  9,  8,  1,  1,  3,  9,  9,  7,  3,  4,  5,  6,  7,  8,  9,  6,  8,  7,  8,  5. 

76.  6,  8,  2,  4,  1,  3,  5,  7,  9,  1,  4,  7,  2,  6,  1,  6,  8,  9,  9,  8,  6. 

77.  7,  8,  9,  9,  8,  7,'  6,  5,  4,  4,  5,  6,  3,  8,  7,  7,  8,  3,  7,-  4,  7. 

78.  5,  3,  7,  6,  2,  4,  9,  8,  7,  4,  5,  9,  2,  9,  8,  7,  8,  6,  3,  8,  9. 

79.  2,  7,  5,  3,  9,  6,  8,  8,  8,  5,  6,  2,  5,  6,  8,  3,  4,  7,  6,  6,  9. 
so.  8,  7,  7,  6,  6,  5,  5,  4,  3,  2,  1,  6,  5,  4,  3,  3,  5,  7,  9,  8,  7. 

81.  9,  6,  3,  4,  2,  4,  6,  8,  9,  8,  7,  6,  3,  1,  3,  5,  9,  9, 8,  4,  5. 

82.  4,  3,  5,  8,  7,  6,  5,  1,  3,  2,  4,  5,  7,  8,  2,  4,  5,  7,6,  8,  8. 

83.  3,  5,  7,  2,  2,  4,  6,  7,  8,  3,  6,  9,  9,  8,  7,  6,  3,  2,  5,  1,  7. 

84.  8,  2,  4,  5,  6,  2,  7,  8,  6,  4,  5,  9,  8,  3,  4,  5,  3,  6, 9,  4,  2. 

85.  9,  9,  8,  7,  3,  4,  6,  1,  5,  4,  2,  3,  3,  9,  1,  7,  8,  9,  7,  8,  2. 

86.  8,  5,  6,  9,  4,  3,  3,  4,  6,  1,  4,  3,  1,  6,  9,  9,  6,  2,  3,  2,  4. 

87.  8,  2,  4,  8,  1,  5,  1,  5,  8,  3,  3,  1,  9,  5,  4,  8,  6,  3,  5,  8,  9. 

88.  1,  5,  8,  6,  3,  5, 1,  9,  3,  3,  7,  7,  2,  5,  9,  2,  8,  6,  6,  2,  2. 

Ex.  74.  3,  10,  16,  24,  33,  35,  etc.    Allow  9  seconds  for  each  example. 

89.  On  the  next  page  find  the  sum  of  the  columns  giving 
the  population  of  the  U.  S.  in  1850. 

90.  In  the  same  manner  find  the  sum  of  the  columns  for 
1860. 

91.  Find  the  sum  of  the  columns  for  1870. 

92.  Find  the  sum  of  the  columns  for  1880. 

93.  Find  the  sum  of  the  columns  for  1890. 

Ex.  89.   9,  16,  20,  25,  31,  .  .  .   166  ;   5,  13,  22,  24,  28,  .  .  .  171 ;  etc. 
Allow  19  seconds  for  a  column. 


ADDITION. 

13 

U.S. 

1850 

1860 

1870 

1880 

1890 

N.Y. 

3,097,394 

3,880,735 

4,382,759 

5,082,871 

5,997,853 

Penn. 

2,311,786 

2,906,215 

3,521,951 

4,282,891 

5,258,014 

111. 

851,470 

1,711,951 

2,539,891 

3,077,871 

3,826,351 

Ohio 

1,980,329 

2,339,511 

2,665,260 

3,198,062 

3,672,316 

Mo. 

682,044 

.  1,182,012 

1,721,295 

2,168,380 

2,679,184 

Mass. 

994,514 

1,231,066 

1,457,351 

1,783,085 

2,238,943 

Tex. 

212,592 

604,215 

818,579 

1,591,749 

2,235,523 

Ind. 

988,416 

1,350,428 

1,680,637 

1,978,301 

2,192,404 

Mich. 

397,654 

749,113 

1,184,059 

1,636,937 

2,093,889 

Iowa 

192,214 

674,913 

1,194,020 

1,624,615 

1,911,896 

Ky. 

982,405 

1,155,684 

1,321,011 

1,648,690 

1,858,635 

Ga. 

906,185 

1,057,286 

1,184,109 

1,542,180 

1,837,353 

Tenn. 

1,002,717 

1,109,801 

1,258,520 

1,542,359 

1,767,518 

Wis. 

305,391 

775,881 

1,054,670 

1,315,491 

1,686,880 

Va.    * 

1,421,661 

1,596,318 

1,225,163 

1,512,565 

1,655,980 

N.C. 

869,039 

992,622 

1,071,361 

1,399,750 

1,617,947 

Ala. 

771,623 

964,201 

996,992 

1,262,505 

1,513,017 

N.J. 

489,555 

672,035 

906,096 

1,131,116 

1,444,933 

Kans. 

107,206 

364,399 

996,096 

1,427,096 

Minn. 

6,077 

172,023 

439,706 

780,773 

1,301,826 

Miss. 

606,526 

791,305 

827,922 

1,131,597 

1,286,600 

Cal. 

92,597 

379,994 

560,247 

864,694 

1,208,130 

S.C. 

668,507 

703,708 

705,606 

995,577 

1,151,149 

Ark. 

209,897 

435,450 

484,471 

802,525 

1,128,179 

La. 

517,762 

708,002 

726,915 

939,946 

1,118,587 

Nebr. 

28,841 

122,993 

452,402 

1,058,910 

Md. 

583,034 

687,049 

780,894 

934,943 

1,042,390 

W.Va. 

442,014 

618,457 

762,794 

Conn. 

370,792 

460,147 

537,454 

622,700 

746,258 

Me. 

583,169 

628,279 

626,915 

648,936 

661,086 

Colo. 

34,277 

39,864 

194,327 

412,198 

Fla. 

87,445 

140,424 

187,748 

269,493 

391,422 

N.H. 

317,976 

326,073 

318,300 

346,991 

376,530 

R.I. 

147,545 

174,620 

217,353 

276,531 

345,506 

Vt. 

314,120 

315,098 

330,551 

332,286 

332,422 

Oreg. 

13,294 

52,465 

90,923 

174,768 

313,767 

D.C, 

51,687 

75,080 

131,700 

177,624 

230,392 

Dei. 

91,532 

112,216 

125,015 

146,608 

168,493 

Nev. 

6,857 

42,491 

62,266 

45,761 

Best 

72,927 

184,497 

311,030 

606,819 

1,621,118 

14  AMERICAN   MENTAL   ARITHMETIC. 

§  3.     Combinations  —  Three  Figures. 
Declare  the  sums : 

01888757461111374922 

94.  9,  7,  8,  7,  3,  3,  4,  8,  4,  7,  1,  1,  5,  7,  8,  7,  9,  9,  6,  8. 

29223139588999979999 


95. 


96. 


97. 


98. 


99. 


3 

1 

5 

8 

C» 

3 

2 

1 

1 

2 

5 

6 

8 

8 

8 

5 

9 

1 

1 

1 

7, 

7, 

r>. 

(A 

6, 

4, 

2, 

4, 

6, 

5, 

7, 

7, 

8, 

8, 

8, 

5, 

9, 

4, 

8,9. 

8 

9 

9 

9 

9 

4 

6 

8 

8 

9 

8 

9 

8 

9 

5 

3 

3 

9 

9 

9 

4 

5 

9 

8 

3 

L> 

2 

2 

2 

3 

3 

3 

4 

4 

6 

3 

5 

4 

6 

5 

8, 

6, 

6, 

8, 

5, 

5, 

2, 

3, 

5, 

5, 

3, 

9, 

4, 

8, 

8, 

4, 

9, 

4, 

9, 

8. 

<s 

8 

5 

3 

9 

8 

9 

7 

5 

3 

4 

9 

4 

9 

9 

6 

9 

7 

9 

9 

3 

7 

4 

6 

7 

4 

7  2 

7 

1 

1 

1 

3 

4 

4 

4 

3 

4 

4 

5 

3, 

8, 

5, 

8, 

8, 

4, 

7, 

4, 

9, 

6, 

5, 

4, 

7, 

5, 

6, 

6, 

5, 

7, 

5, 

7. 

(5 

7 

5 

8 

8 

6 

9 

9 

9 

6 

6 

5 

9 

9 

9 

8 

8 

8 

8 

7 

3 

1 

6 

6 

1 

6 

2 

3 

1 

1 

1 

4 

1 

1 

3 

4 

5 

2 

2 

1 

6, 

1, 

6, 

7, 

1, 

6, 

3, 

8, 

1, 

8, 

2, 

4, 

1, 

1, 

3, 

C>, 

6, 

2, 

2, 

3. 

6 

1 

7 

7 

2 

8 

3 

9 

3 

8 

2 

9 

4 

5 

3 

6 

6 

3 

4 

3 

3 

1 

3 

5 

2 

1 

2 

1 

1 

3 

1 

4 

1 

5 

2 

5 

1 

6 

1 

4 

7, 

3, 

6, 

6, 

3, 

2, 

2, 

2, 

2, 

6, 

2, 

6, 

4, 

5, 

2, 

5, 

3, 

6, 

1. 

7. 

7 

4 

9 

7 

4 

4 

2 

6 

3 

8 

5 

7 

4 

7 

5 

8 

5 

6 

7 

7 

51432222111211222112 

100.  5,  1,  5,  5,  6,  5,  4,  4,  2,  4,  3,  6,  2,  7,  4,  7,  2,  6,  3,  3. 
66776664876  8^  88778985 
22117233214  2  3  3313114 

101.  4,  6,  2,  3,  7,  5,  3,  3,  3,  5,  3,  3,  4,  6,  4,  6,  5,  4,  5,  5. 
879917979778  8  7  9  7  5  9  8  6 
7464  5  978733  49  8  5  9  6  3  4  8 

102.  7,  8,  6,  3,  2,  9,  6,  9,  4,  8,  4,  7,  2,  3,  4,  7,  6,  4,  5,  5. 
89718431279814926347 
16892585232796378799 

103.  4,  7,  2,  8,  9,  4,  7,  4,  1,  4,  4,  4,  5,  4,  3,  1.  9,  4,  6,  5. 
58177363932789892134 

9         9 
Ex.  94.  20,  17,  18,  17,  14,  11,  12,  etc.   Look  upon  q  as  18  ;  then  9 

»> 

18 
appears  2  ;  say  20.    Speak  no  words  except  the  sums.    Allow  14  seconds 

for  each  example. 


ADDITION.  15 

§  4.    Combinations  —  Four  Figures. 

Read  the  sums : 

99999999999999999999 
„^  99999999999999999999 
104'  9'  5'  6'  2'  5'  V  9>  5'  2'  8'  8'  6'  8'  8'  V  6'  4'  7'  8'  4' 

94625621179223132454 


9  9  9  9  9  9 

9  9 

9  9 

9  9  9 

9  9  9  9  9  9  9 

8  8  8  8  8  8 

8  8 

8  8 

8  8  8 

8  8  8  8  8  8  8 

105. 

V  5'  9'  2'  7'  6' 

4'  3' 

V  6' 

4'  7'  8' 

6'  8>  5'  9'  8'  V  9* 

15  7  17  4 

4  3 

5  2 

113 

16  2  3  7  2  4 

12  2  3  3  4 

5  3 

4  9 

8  7  6 

9  8  7  6  9  8  7 

112  2  3  3 

5  1 

1  2 

3  4  5 

3  4  5  6  4  5  6 

106. 

9'  9'  8'  5'  8'  8' 

6'  8' 

7'  9' 

4'  2'  4' 

6'  6'  7'  5'  3'  8'  5* 

9  8  8  5  8  7 

6  3 

5  1 

12  3 

4  12  2  113 

9  8  9  8  9  9 

9  8 

7  6 

5  8  7 

6  5  7  6  5  4  6 

6  7  7  8  8  9 

1  2 

3  4 

5  12 

3  4  12  3  4  1 

107. 

V  4'  3'  4'  9'  8' 

1'  6' 

V  8' 

9'  8'  7' 

6'  2'  6'  6'  V  8'  8* 

3  2  2  4  11 

1  6 

6  5 

4  5  7 

5  15  14  18 

12  3  3  6  1 

5  9 

4  8 

3  7  2 

6  2  19  7  5  3 

8  9  6  5  8  2 

6  1 

5  9 

4  8  3 

7  4  3  19  7  5 

108. 

5'  6'  7'  8'  9'  3' 

V  2> 

6'  1' 

5'  9'  4> 

8?  6'  5'  3'  V  9'  7' 

3  6  9  7  5  4 

8  3 

7  2 

6  15 

9  8  7  5  3  19 

3  8  4  6  6  8 

8  7 

8  9 

7  9  5 

6  8  3  4  9  3  9 

6  4  8  2  5  2 

5  7 

8  1 

2  4  4 

7  5  3  4  9  8  6 

109. 

2'  1'  2'  8'  4'  7' 

2'  7' 

8>  8' 

8'  V  8' 

2'  9'  6'  8'  2'  7?  2' 

5  7  6  4  7  3 

5  7 

8  3 

3  6  3 

12  6  8  15  1 

7  7  7  7  7  7 

7  7 

7  7 

7  7  7 

7  7  7  7  7  7  7 

6  6  5  5  6  6 

5  4 

3  2 

16  6 

5  5  4  3  2  18 

110. 

1'  2'  3'  4'  5'  6' 

7>  8' 

9'  5' 

4'  3'  2' 

V  9'  5'  6'  V  8'  9" 

9  8  7  6  5  4 

3  2 

1  9 

8  7  6 

5  4  3  2  19  8 

Ex.  104.  36,  27,  30,  22,  28,  31,  29,  etc.  Look  upon  g  as  18;  then 

9 

g  appears  as  :S ;  say  36.     Speak  no  word  except  the   sums.      Allow  18 

9 

seconds  for  each  example. 


16  AMERICAN  MENTAL   ARITHMETIC. 

§  5.     Addends. 
Add: 

111.  18, 17,  11, 16,  15,  14,  13,  12,  19,  20,  17,  19, 18, 17, 19. 

112.  19,  19,  18,  18,  17,  17,  16,  16,  15,  15,  14,  14, 13, 13, 12. 

113.  21,  22,  23, 16,  15,  18,  11,  10,  19,  24,  16,  18, 12, 14, 15. 

114.  11,  12,  13,  14,  15,  16,  17,  18,  19,  20,  21,  22,  23,  24,  25. 
lis.  26,  27,  28,  29,  30,  31,  32,  33,  34,  35,  36,  37,  38,  39,  40. 

116.  39,  38,  37,  36,  35,  34,  33,  32,  31,  30,  29,  28,  27,  26,  25. 

117.  24,  23,  22,  21,  20,  19,  18,  17,  16,  15,  14,  13, 12, 11, 10. 

118.  40,  11,  39,  12,  38, 13,  37,  14,  36,  15,  35,  16,  34, 17,  33. 

Ex.  111.  35,  46,  62,  77,  91,  etc. 

A  glance  determines  whether  the  sum  of  the  units  is  more 

than  9  or  less  than  10.     If  more  than  9,  we  increase  the  sum 

of  the  tens  by  1 ;  if  less  than  10,  we  take  sum  of  the  tens. 

Ex.  111.  18  +  17  ;  the  sum  of  the  units  is  more  than  10  ;  we  increase 
the  sum  of  the  tens  by  1  and  say  35  ;  35  +  11 ;  the  sum  of  the  units  is  less 
than  9  ;  we  take  the  sum  of  the  tens  and  say  46,  etc. 

Find  the  sum : 

119.  78  +  94.  122.  74  +  92.  125.  63  +  18. 

120.  86  +  75.  123.  83  +  75.  126.  94  +  87. 

121.  73  +  68.  124.  43  +  58.  127.  63  +  25. 

Ex.  119.   172. 

Beginning  with  1  count  to  about  200 : 

128.  By  19.  131.    By  16.  134.    By  13. 

129.  By  18.  132.    By  15.  135.   By  12. 

130.  By  17.  133.    By  14.  136.    By  11. 
Ex.  128.   1,  20,  39,  58,  77,  96,  etc. 


ADDITION.  17 

Add: 

''        2469477731  ...    7542468273 

137.  r  f  5,  5,  2,  2,   f  6,  g,   9-  141.  x,  5,  4,   6,  4,  9,  4,  g,  g,  g. 

,00  3944969726  _._  1111289986 

138.  2,  g,  2,  7,  g,  g,  g,  2,  g,  3.  142.  g,  g,.  g,  9,  4,  r   g,  4,  2,  3- 

1M  5  2  5  4  U  5  5  6  7  ...  4653745689 

139-  9'  6'  t   9'  1*  f   5'  5'  6'  3'  143*  1'  2'  4'  5'  4'  3'  5'  T  8'  6* 

,An  1377392611  1AA  2722112468 

140.  2,  ^  4,  7,  3,  7,  g,  6,  g,  g.  144.  6,  2,  8,  4,  1,  7,  3,  g,  7,  g. 


6391462135  2417232157 

145.  7,  8,  7,  9,  8,  3,  1,  1,  4,  8.  148.  8,  6,  2,  4,  3,  6,  8,  9,  2,  2. 
754725  5  863  4428928649 

1924538111  2548324554 

146.  5,   6,  7,  7,  4,  2,  1,  8,  2,  1.  149.  6,  2,  6,  7,  5,  1,  2,  6,  7,  9. 
9827492816  4548897294 

3446937111  5725644127 

147.  5,  8,  4,  1,  6,  1,  9,  5,  9,  5.  150.  2,  8,  7,  4,  4,  5,  1,  7,  1,  2. 
1345221116  1766914895 


2694923647     3132322133 

,-,  1417975952  __,  2122278868 

151'  9'  5'  5'  1'  9*  6'  2'  5'  5'  8*  153*  4'  3'  3'  8'  6'  9'  5'  8'  2'  2* 

3519261793     1984865957 

5892953941     4175357282 

-„  9786365882  ncA  8166755514 

152'  7'  3'  4'  4'  7'  1'  6'  2'  7'  5*  154'  6'  6'  7'  2'  4'  6'  9'  9'  1?  7* 

4399931257     9833767115 

Ex.  137.  9,  20,  31,  45,  51,  60,  74,  etc,  Look  at  J,  say  9 ;  at  4  20 ;  at  ®, 
31 ;  at  k,  45 ;  etc.  4  17 

Ex.  154.  27,  43,  66,  82,  etc.    Look  at  5,  say  27  ;  at  g,  43  ;  at  fj,  66 ;  etc. 

9  8  3 

By  reading  several  figures  at  a  glance  and  combining  as 
in  this  section,  the  columns  on  page  13  may  be  added  in  5 
seconds  each. 

AM.    MENT.    AR.  — 2 


18  AMERICAN  MENTAL   ARITHMETIC. 

§  6.    Problems. 

Declare  the  answer  to  each  as  quickly  as  possible  with- 
out reading  the  problem  aloud  and  before  explaining. 

If  required  to  explain,  avoid  repetitions  and  unneces- 
sary words. 

155.  A  paid  16^  for  a  book,  15?  Ans,  36^  He  paid  in 
for  a  slate,  and  5^  for  a  pencil ;  how  all  the  sum  of  16^,  15^,  and 
much  did  he  pay  in  all  ?     Explain.  ^  or  36^ 

156.  Jane  bought  some  apples  for  10^,  some  peaches  for 
18^,  some  plums  for  20^,  and  an  orange  for  8^;  how  much 
did  she  pay  for  all  ? 

157.  John  has  28  marbles  in  one  bag,  16  in  another,  14  in 
another,  35  in  another,  and  19  in  another;  how  many  has 
he  in  all  ? 

158.  There  are  9  birds  in  one  flock  and  27  in  another  ; 
how  many  are  there  in  both  ?     Explain. 

159.  One  day  I  walked  5  miles  and  the  next  day  14  miles ; 
how  far  did  I  walk  in  both  days  ? 

160.  A  man  has  horses  in  3  pastures :  in  the  first  9,  in  the 
second  11,  and  in  the  third  13;  how  many  has  he  in  all? 
Explain. 

161.  A  baker  sold  36  loaves  of  bread  on  Monday,  30  on 
Tuesday,  27  on  Wednesday,  34  on  Thursday,  25  on  Friday, 
and  40  on  Saturday ;  how  many  loaves  did  he  sell  during  the 
week? 

162.  A  man  has  5  baskets  of  eggs :  in  the  first  basket  there 
are  24  eggs,  in  the  second  36,  in  the  third  18,  in  the  foarth 
12,  and  in  the  fifth  16 ;  how  many  has  he  in  the  five  bas- 
kets? 


ADDITION.  19 

163.  38  years,  29  years,  10  years,  19  years,  23  years,  and 
14  years  are  how  many  years  in  all  ? 

164.  Walter  saw  three  flocks  of  prairie  chickens ;  the  first 
contained  28  chickens,  the  second  19,  and  the  third  33; 
how  many  chickens  did  he  see  ? 

165.  I  bought  on  account :  a  cabbage  for  10  cents,  a  dozen 
eggs  for  24  cents,  a  peck  of  apples  for  15  cents,  a  bushel  of 
potatoes  for  65  cents,  and  a  quart  of  beans  for  20  cents  ;  how 
much  do  I  owe  the  merchant  for  these  ? 

166.  How  many  books  are  there  in  the  Bible  if  the  Old 
Testament  contains  39  and  the  New  Testament  27  books  ? 

167.  In  a  factory  there  are  15  men,  12  women,  17  girls, 
and  21  boys  at  work ;  how  many  persons  are  employed  in 
the  factory? 

168.  Alfred  earned  44  cents  one  day,  50  cents  the  next 
day,  and  found  35  cents ;  how  many  cents  did  he  then  have  ? 

169.  Clay  studied  25  minutes  one  afternoon,  55  minutes 
the  next  afternoon,  and  12  minutes  the  next  afternoon; 
how  many  minutes  did  he  study  in  all? 

170.  A  farmer  raised  36  bushels  of  wheat,  18  bushels  of 
oats,  27  bushels  of  rye,  and  19  bushels  of  corn ;  how  many 
bushels  of  grain  did  he  raise  ? 

171.  A  lady  canned,  during  the  summer,  12  quarts  of 
peaches,  9  quarts  of  cherries,  26  quarts  of  strawberries, 
17  quarts  of  blackberries,  13  quarts  of  raspberries,  and  19 
quarts  of  gooseberries ;  how  many  quarts  of  fruit  has  she 
for  winter  use? 

172.  Find  the  number  of  days  in  the  first  six  months  of 
the  year  when  January  has  31  days,  February  29  days, 
March  31  days,  April  30  days,  May  31  days,  and  June 
30  days. 


20  AMERICAN  MENTAL  ARITHMETIC. 

173.  Find  the  sum  of  62,  15,  9,  32,  27,  18,  8,  6,  and  4. 

174.  A  carpenter  used  7  bunches  of  lath  for  the  kitchen, 
16  bunches  for  the  dining-room,  12  bunches  for  the  parlor, 
and  11  bunches  for  a  bed-room ;  how  many  bunches  did  he 
use  for  the  four  rooms? 

175.  Bought  berries  for  17  cents,  cherries  for  19  cents, 
and  apples  for  13  cents ;  what  was  the  cost  of  all? 

176.  A  lady  bought  a  dress  for  $18,  a  muff  for  $16,  a 
shawl  for  $17,  and  other  articles  for  $19;  what  was  the 
whole  bill? 

177.  A  merchant  sold  18  barrels  of  flour  one  week,  16  the 
next  week,  12  the  next,  13  the  next,  and  14  the  next;  how 
many  barrels  did  he  sell  during  the  five  weeks  ? 

178.  Mary  had  56  oranges,  and  Susan  had  19  more  than 
Mary ;  how  many  had  Susan  ? 

179.  A  man  is  48  years  old,  and  his  wife  is  36  years  old : 
what  is  the  sum  of  their  ages  ? 

180.  James  had  59  cents  and  found  48  cents ;  how  many 
cents  had  he  then  ? 

181.  Mary  gave  58  cents  to  her  brother  and  96  cents  to 
her  sister ;  how  many  cents  did  she  give  away  ? 

182.  A  merchant  sold  rice  for  $198,  sugar  for  $18,  oil  for 
$17,  candy  for  $13,  molasses  for  $16,  and  salt  for  $12;  how 
much  did  he  receive  for  all? 

183.  A  girl  made  15  red  pin-wheels,  16  blue  ones,  and  17 
blue  and  white.     How  many  pin- wheels  did  she  have  ? 

184.  It  is  38  miles  from  A  to  B,  19  miles  from  B  to  C,  17 
miles  from  C  to  D,  18  miles  from  D  to  E ;  how  many  miles 
does  a  man  travel  who  goes  from  A  to  E,  passing  through 
B,  C,  andD? 


SUBTRACTION. 


Subtraction  is  indicated  by  the 
sign  -. 

The  number  to  be  subtracted  is 
the  subtrahend. 

The  number  from  which  to  sub- 
tract is  the  minuend. 

The  result  is  the  difference  or 
remainder. 

The  difference  between  any  two 
numbers  may  be  found  by  counting. 

Subtraction  is  a  process  shorter 
than  counting  for  finding  the 
difference  between  numbers. 

Read  the  remainders  as  rapidly  as 
1.   10   10   10   10   10   10   10   10 


Illustration. 
8-3  =  5 
read 
8  minus  3  equals  5. 
8,  minuend. 
3,  subtrahend. 
5,  difference  or  remainder. 

To  find  the  difference  be- 
tween 8  and  3  by  counting. 

Counting  to  8  and  making  a 
mark  at  each  count,  ////////; 
counting  to  3  and  crossing  a 
mark  at  each  count,  XXX/////; 
counting  what  is  left,  we 
have  5. 

possible : 

10   11   11  11  11  11  11 


9  5  3 

7 

4  18 

2  6  9 

6 

2 

4  8  3 

2.  11  11  12 

12 

12  12  12 

12  12  13 

13 

13 

13  13  13 

5  7  9 

5 

8  4  6 

3  7   9 

5 

7 

4  8  6 

3.  14  14  14 

14 

14  15  15 

15  15  16 

16 

16 

17  17  18 

9  7  5 

8 

6  9  7 

8  6   9 

7 

8 

9  8  9 

Ex.  1.  1,  5,  7,  3,  6,  9,  etc.    Do  not  say  9  from  10  leaves  1. 


21 


22 


AMERICAN  MENTAL   ARITHMETIC. 


What  must  be  added  to  : 

4.  9  to  make  18  ?  7.    8,  9,  7,  6  to  make  15  ? 

5.  9,  8  to  make  17  ?  8.    7,  9,  5,  6,  8  to  make  14  ? 

6.  7,  9,  8  to  make  16  ?  9.   5,  8,  7,  9,  6,  4  to  make  13  ? 

io.   6,  7,  5,4,  8,  3,  9  to  make  12? 

11.  5,  8,  7,  4,  3,  6,  2,  9  to  make  11  ? 

12.  9,  2,  3,  5,  4,  7,  6,  1,  8  to  make  10? 

Ex.  8.   7,  5,  9,  8,  6. 

Beginning  with  100  count  backwards : 

13.  By  9.  16.    By  6.  19.    By  3. 

14.  By  8.  17.    By  5.  20.    By  2. 

15.  By  7.  18.   By  4.  21.   By  1. 

Ex.  13.    100,  91,  82,  etc. 


Begin  at  the  right  and  read  the  remainder  : 


22. 

98736 
86224 

26. 

85672 
43461 


23. 

5897638 
2634521 

27. 

5463267 
3252145 


24. 

12345678 
1343527 

28. 

76924711 
56412300 


25. 

98768975 
82345723 

29. 

64723108 
23412106 


Ex.  22.  Say  2,  1,  5,  2,  1.  The  habit  of  saying  "4  from  6  leaves  2  ;  2 
from  3  leaves  1 ;  2  from  7  leaves  5,"  etc.,  should  be  broken  up.  While  the 
four  words,  "  4  from  6  leaves,1'  are  being  formed,  no  progress  can  be  made  in 
subtracting.  There  is  no  reason  why  the  student  should  not  call  off  the 
figures  of  the  remainder  as  rapidly  as  he  can  talk. 


SUBTRACTION. 
Begin  at  the  right  and  read  the  remainder 


23 


30.            31. 

32 

33. 

36854       3000205 
29876       1864783 

43000005       568120001 
17652436       497203854 

34. 

35. 

36. 

503784325 

282958298 

6234567890 
3929802958 

3050702003 

1234567898 

37. 

38. 

286540302567200 
192830605088739 

806304205102030 
507080610503028 

39. 

40. 

736904521300671 
497000457369712 

192470030060091 
141398765432189 

41. 

42. 

862300100439610 

765489012786937 

572000000700123 
324986574309165 

43. 

44. 

432176090135200 
168349827563425 

678230004000500 
437654321234567 

Ex.  30.   Say  8,  7,  9,  6,  and  no  other  words.    Practice  will  enable  the 
student  to  read  the  figures  of  the  remainder  almost  as  rapidly  as  he  can  talk. 


24 


AMERICAN  MENTAL   ARITHMETIC. 


45.  To  make  100,  what  must  be  added  to  56?  48?  32? 
74?  83?  76?  25?  73?  44?  86?  92?  38?  53?  27?  49?  33? 
18?  58?  67?  77? 

46.  To  make  1000,  what  must  be  added  to  72?  102?  148? 
156?  63?  179?  185?  196?  144?  156?  175?  183?  122?  104? 
157?  163?  177?  192? 

47.  To  make  1000,  what  must  be  added  to  676?  687?  575? 
762?  349?  534?  296?  105?  428?  777?  388?  499? 

48.  To  make  1000,  what  must  be  added  to  375  ?  804?  783? 
926?  439?  604?  593?  355?  707?  599? 

49.  To  make  10000,  what  must  be  added  to  3608?  5732? 
4963?  6078?  7095?  2801?  5678?  4209? 

50.  From  1000000,  subtract  886097,  407864,  360835, 
479632,  582769,  380803,  760967,  320457,  978654. 

51.  From  100000000,  subtract  23456789,  92037405, 
50640720,  40009265,  70904055,  66090207. 


52. 

1000000000 
372840625 


53. 

1000000000 
102030458 


54. 

1000000000 
289076430 


55. 

1000000000 
203572763 


56. 

1000000000 
123456789 


57. 

1000000000 
246813579 


58. 

1000000000 
764031246 


59. 

1000000000 
837964012 


60. 

1000000000 
543212345 


Ex.  48.  625,  196,  217,  etc.  In  these  examples,  it  is  best  to  begin  at  the 
left  and  call  ont  what  must  be  added  to  each  figure  of  the  subtrahend  except 
the  last  to  make  0,  but  what  must  be  added  to  the  last  figure  to  make  10. 
The  student  should  read  results  as  fast  as  he  can  talk. 


SUBTRACTION.  25 

61.  From  276  take  189.  72.  From  506  take  489. 

62.  From  364  take  278.  73.  From  287  take  198. 

63.  From  467  take  389.  74.  From  802  take  746. 

64.  From  123  take  74.  75.  Take  123  from  210. 

65.  From  106  take  98.  76.  Take  299  from  323. 

66.  From  207  take  199.  77.  Take  145  from  223. 

67.  From  111  take  46.  78.  Take  345  from  421. 

68.  From  203  take  145.  79.  Take  258  from  324. 

69.  From  209  take  167.  80.  Take  456  from  531. 

70.  From  245  take  169.  81.  Take  489  from  503. 

71.  From  456  take  389.  82.  Take  286  from  345. 

Ex.  61.  87.  To  make  200,  11  must  be  added  to  189  ;  76  +  11  =  87.  Do 
not  say,  "9  from  16  leaves  7  ;  9  from  17  leaves  8." 

83.  To  make  922,  what  must  be  added  to  648?  396?  479? 
553?  764?  875?  283?  697?  785?  892?  189?  527?  819? 
634?  311?  439?   510?   609? 

84.  To  make  816,  what  must  be  added  to  378?  496?  785? 
396?  519?  439?  382?  786?  758?  729?  715?  638?  525? 
444?   775?   314?   678?   248? 

85.  To  make  725,  what  must  be  added  to  648  ?  639?  675? 
686?  695?  683?  681?  649?  663?  671?  535?  598?  419? 
307?   212?   199?   25?   63? 

86.  To  make  513,  what  must  be  added  to  416?  438?  269? 
183?  68?  75?  233?  175?  254?  285?  54?  19?  153?  240? 
369?  452?  387?   299? 

Ex.  83.  274,  526,  443,  etc.  Consider  what  must  be  added  to  each  to 
make  900;  then  add  22;  e.g.  to  make  900,  252  must  be  added  to  648; 
252  +  22  =  274  ;  to  make  900,  504  must  be  added  to  396  ;  504  +  22  =  526, 
etc. 


26  AMERICAN  MENTAL  ARITHMETIC. 

§  7.    Problems. 

Declare  the  answers  to  each  as  quickly  as  possible  with* 
oat  reading  the  problem  aloud  and  before  explaining. 

If  required  to  explain,  avoid  repetitions  and  unneces- 
sary words. 

87.  From  a  box  containing  Ans.  8  marbles.  There  remained 
37  marbles  29  were  lost ;  how  tlie  difference  between  37  marbles 
many  remained  ?     Explain.         and  29  marbles'  or  8  marbles- 

88.  Henry  had  75^  and  gave  45^  for  a  knife  ;  how  many 
had  he  left? 

89.  A  man  had  $  145  in  the  bank  and  drew  out  $  89;  how 
many  dollars  had  he  left  in  the  bank  ?     Explain. 

90.  A  merchant  bought  124  barrels  of  apples,  and  sold  98 
barrels  ;  how  many  had  he  left  ? 

91.  A  farmer  had  115  sheep  and  after  selling  some  he  had 
86  left ;  how  many  did  he  sell  ?     Explain. 

92.  In  an  orchard  there  are  100  peach  trees  and  57  plum 
trees  ;  how  many  more  peach  trees  than  plums  trees  are  there  ? 

93.  A  receives  a  salary  of  $125  per  month,  and  after  pay- 
ing his  necessary  expenses  he  has  $57  left;  what  are  his 
expenses  ?     Explain. 

94.  B  borrowed  $105  and  paid  $  79  of  the  debt ;  how  much 
did  he  still  owe  ? 

95.  The  sum  of  two  numbers  is  122,  one  of  the  numbers  is 
39;  what  is  the  other?     Explain. 

96.  A  pays  $130  for  a  horse  and  a  saddle.  He  pays  $13 
for  the  saddle ;  how  much  does  he  pay  for  the  horse  ? 

97.  From  a  bin  containing  112  bushels  of  corn,  76  bushels 
were  sold ;  how  many  bushels  remained  ? 


SUBTRACTION.  27 

98.  From  a  school  of  109  pupils,  18  were  absent ;  how  many 
were  present?     Explain. 

99.  Ray's  father  gave  him  75^,  his  mother  gave  him  45^, 
his  sister  gave  him  20^,  and  his  brother  gave  him  15^.  He 
spent  92^ ;  how  much  had  he  left  ?     Explain. 

100.  The  sum  of  three  numbers  is  145 ;  the  first  is  24,  the 
second  is  48 ;  what  is  the  third  ? 

101.  A  merchant  had  on  hand  36  pounds  of  butter ;  he 
bought  95  pounds  more  and  then  sold  57  pounds ;  how 
much  had  he  left  ? 

102.  What  is  the  difference  between  69  +  82  and  73  +  49? 

103.  What  is  the  difference  between  132  -  29  and  110  -  61  ? 

104.  What  is  the  difference  between  101  +  48  +  24  and 
111-19? 

105.  What  number  subtracted  from  98  +  23  will  leave  73? 

106.  Ethel  had  45^ ;  she  spent  29^,  after  which  she  earned 
57^ ;  how  many  cents  had  she  then  ? 

107.  A  farmer  raised  150  bushels  of  wheat ;  he  sold  at  one 
time  35  bushels,  at  another  time  49  bushels  and  kept  the 
remainder ;  how  many  bushels  did  he  keep  ? 

108.  A  earns  $  100  per  month,  and  pays  $  15  for  board  and 
$46  for  other  expenses ;  how  much  does  he  save  each  month? 

109.  From  a  piece  of  carpet  containing  76  yards,  two 
pieces  were  cut  and  29  yards  remained ;  the  first  piece  cut  off 
contained  13  yards;  how  many  yards  did  the  second  piece 
contain  ?     Explain. 

110.  Subtract  87  from  104,  add  45  to  the  remainder,  and 
declare  the  result. 

ill.    On  Monday  a  gentleman  deposited  in  a  bank  $53, 
on  Tuesday  he  deposited  $85,  on  Wednesday  he  drew  out 
how  much  did  he  leave  in  the  bank  ? 


28  AMERICAN  MENTAL  ARITHMETIC. 

112.  A  owes  me  76^;  I  owe  him  91^;  how  may  we  settle 
the  account  ?     Explain. 

113.  From  the  sum  of  65  and  88  subtract  the  sum  of  37 
and  49. 

114.  James  bought  100  oranges  and  sold  68  of  them ;  how 
many  had  he  left  ? 

115.  John  is  77  years  old,  and  Joseph  is  38 ;  John  is  how 
many  years  older  than  Joseph  ? 

116.  In  a  school  of  66  pupils  29  are  present;  how  many 
are  absent  ? 

117.  Of  1000  men  128  were  sick ;  how  many  were  well  ? 

118.  From  a  herd  of  256  cattle  189  were  sold  ;  how  many 
remained? 

119.  A  man  sold  48  cows,  then  bought  19,  and  then  had 
65 ;  how  many  had  he  at  first  ? 

120.  Prove  that  your  answer  to  the  119th  is  correct. 

121.  A  farmer  exchanged  eggs  costing  35^,  butter  19^, 
potatoes  $1.05,  for  cloth  costing  36^,  sugar  50^,  starch  12^; 
how  much  was  due  him  ? 

122.  A  man  bought  a  horse  for  $57,  received  for  his  use 
$19,  and  paid  for  his  keeping  $12 ;  he  sold  him  for  $65 ;  how 
much  did  he  gain  ? 

123.  Paid  $37  for  sugar,  $29  for  molasses  ;  how  much  did 
both  cost?  How  much  more  did  the  sugar  cost  than  the 
molasses  ? 

124.  John  bought  35  apples  at  one  store  and  48  at  another ; 
he  sold  29  of  them ;  how  many  remained  ? 

125.  I  bought  a  horse  for  $  65 ;  for  how  much  must  I  sell 
him  to  gain  $38? 

126.  A  farmer  sold  a  cow  for  $38,  which  was  $19  more 
than  she  cost  him ;  how  much  did  he  pay  for  her  ? 


MULTIPLICATION. 


Multiplication  is  indicated  by  the 
sign  x . 

The  number  to  be  multiplied  is  the 
multiplicand. 

The  number  by  which  to  multiply 
is  the  multiplier. 

The  result  is  the  product. 

Any  number  of  times  a  given  num- 
ber may  be  found  by  adding. 

Multiplication  is  a  process  shorter 
than  adding  for  finding  the  sum 
of  equal  addends. 


Illustration. 

6  x  4  =  24 

read 

6  multiplied  by  4  equals  24 

or 

4  times  6  equals  24. 

6,  multiplicand. 

4,  multiplier. 

24,  product. 

To  multiply  6  by  4  by 
adding. 

6x4=6+6+6+6 
or 
6  taken  4  times  as  an  ad- 
dend. 
.-.  6  x  4  =  24. 


X 

1 

2  |  3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

12 

12 

24  36 

48 

60 

72 

84 

96 

108 

120 

132 

144 

11 

11 

22 

33 

44 

55 

66 

77 

88 

99 
90 

110 
100 

121 
110 

132 
120 

10 

10 

20 

30 

40 

50 

60 

70 

80 

9 

9 

18 

27 

36 

45 

54 

63 

72 

81 

90 

99 

108 

8 

8 

16 

24 

32 

40 

48 

56 

64 

72 

80 

88 

96 

7 
6 

7 
6 

14 
12 

21 
18 

28 
24 

35 
30 

42 
36 

49 
42 

56 

48 

63 
54 

70 
60 

77 
66 

84 

72 

13  x  13  =  169 

18  x  18  =  324 

24  x  24  =  576 

5  x  5  x  5  =  125 

14  x  14  =  196 

19  x  19  =  361 

25  x  25  =  625 

6  x  6  x  6  =  216 

15  x  15  =  225 

21  x  21  =  441 

2x2x2=8 

7  x  7  x  7  =  343 

16  x  16  =  256 

22  x  22  =  484 

3  x  3  x  3  =  27 

8  x  8  x  8  =  512 

17  x  17  =  289 

23  x  23  =  529 

4  x  4  x  4  =  64 

9  x  9  x  9  =  729 

29 


30  AMERICAN   MENTAL    ARITHMETIC. 

When  a  number  is  multiplied  by  itself,  as  13  x  13,  14  x  14, 
etc.,  the  product  is  called  a  square.  When  the  square  of  a 
number  is  multiplied  by  the  number,  as  2x2x2,  3x3x3, 
etc.,  the  product  is  called  a  cube. 

Declare  the  products  of : 

1.  12,11,10,    9,    8,    7,    6,    5,    4,    3,    2,  by    9. 

2.  9,    6,    2,    5,11,    4,    2,    7,    3,    8,  10,  by    8. 

3.  2,  10,    6,    9,    4,  12,    8,  11,    7,    5,    3,  by    7. 

4.  10,    4,    9,    3,    8,  12,    2,    7,  11,    6,    5,  by    6. 

5.  6,    8,    4,10,    2,    9,12,    3,11,    7,    5,  by    5. 

6.  6,    9,    2,    8,    3,    7,10,12,    4,11,    5,  by    4. 

7.  12,    4,11,    2,10,    6,    9,    2,    8,    5,    7,  by  11. 

8.  11,    2,    7,    3,    2,    4,    6,    5,    8,12,    9,  by  12. 

Ex.  2.   72,  48,  16,  40,  etc.     Do  not  say,  "8  times  9  are  72." 

Declare  the  products  of : 

9.  13  by  1,  2,  3,  4,  5,  6,  7.  15.  19  by  1,  2,  3,  4. 

10.  14  by  1,  2,  3,  4,  5,  6,  7.  16.  21  by  1,  2,  3,  4. 

n.  15  by  1,  2,  3,  4,  5,  6.  17.  22  by  1,  2,  3,  4. 

12.  16  by  1,  2,  3,  4,  5,  6.  18.  23  by  1,  2,  3,  4. 

13.  17  by  1,  2,  3,  4,  5.  19.  24  by  1,  2,  3,  4. 

14.  18  by  1,  2,  3,  4,  5.  20.  25  by  1,  2,  3,  4. 

21.  3  by  33,  27,  31,  28,  26,  29,  30,  32. 

22.  2  by  49,  30,  39,  31,  42,  32,  47,  48,  28,  38,  29,  46,  27,  36, 
26,  41,  33,  45,  43,  37,  34,  40,  35. 

23.  13  x  13,  2x2x2,  14  x  14,  8  x  8  x  8. 

24.  14x14,  9x9x9,  25x25,  4x4x4,  21x21,  7x7x7, 
17  x  17,  5  x  5  x  5,  19  x  19,  6  x  6  x  6,  15  x  15,  3  x  3  x  3,  16  x  16, 
24  x  24. 

Ex.  9.    13,  26,  39,  52,  etc.     Do  not  say,  "  13  times  1  are  13." 


MULTIPLICATION. 


31 


Give  the  multiplication  table  : 

25.  '13  times'  to  13  x  T. 

26.  « 14  times '  to  14  x  7. 

27.  *  15  fo'w08 '  to  15  x  6. 

28.  '  16  times '  to  16  x  6. 

29.  '17  times'  to  17x5. 

30.  '18  times'  to  18x5. 


31.  '19  times'  to  19x5. 

32.  '21  ^'raes  '  to  21  x  4. 

33.  '  22  times '  to  22  x  4. 

34.  '  23  times '  to  23  x  4. 

35.  '  24  times '  to  24  x  4. 

36.  '  25  times '  to  25  x  4. 


Ex.  25.    13  x  1  are  13  ;  13  x  2  are  26 ;  13  x  3  are  39  ;  etc. 

Declare  the  results  rapidly  : 

37.  19x5,    17x4,    16x4,    14x4,  18x5,    19x4,    16x6, 
13x3,  5x16,  17x5,  5x13,  17x3,  4x13,  16x3,  19x3. 

38.  19x2,    3x18,   17x2,    2x16,    15x6,    5x14,    15x5, 
14x3,  13x2,  4x18,  15x3,  2x14,  13x6,  7x14,  2x18. 

39.  4x15,     6x16,     7x13,     9x5x2,     2x3x7,    8x13, 
4  x  14,  6  x  15,  6  x  13,  2  x  6  x  8,  5  x  6  x  2,  6  x  14,  18  x  4,  14  x  6. 

40.  16x5,  15x2,  14x2,  15x4,  14x5,  14x7,  3x4x5, 
2x4x6,  5x4x2,  3x5x6. 

State  rapidly: 

41.  The- squares  of  the  integers  from  1  to  25. 

42.  The  cubes  of  the  integers  from  1  to  9. 

43.  The  square  of  25,  23,  24,  21,  19,  16,  17,  15,  13,  22, 

20,  18. 

44.  The  square  of  12,  14,  16,  18,  20,  22,  24,  13,  15,  17, 
19,  21. 

45.  The  cube  of  9,  6,  3,  1,  4,  7,  8,  5,  2,  5,  7,  9,  4,  6,  8. 

46.  The  cube  of  4,  3,  5,  7,  9,  2,  4,  6,  8, 10,  9,  2,  8,  6,  7. 


32 


AMERICAN  MENTAL   ARITHMETIC. 


Multiply : 

47. 

48. 

49. 

2030876514 
9 

5708392943 

8 

3886546312 

7 

50. 

51. 

52. 

4571263972 
4 

7360925168 
12 

54. 

2784235879 
11 

53. 

55. 

1203045678 

8 

6543712345 

6 

6789098765 
5 

56. 

57. 

58. 

4630902034 
11 

5762198345 

7 

60. 

6290311269 

8 

59. 

61. 

3862048001 
9 

3572607983 
3 

6238496785 
12 

62. 

63. 

64. 

4837254912 
4 

5789564328 
6 

7912765046 
11 

Ex.  47.  36,  12,  40,  58,  68,  etc.  Do  not  say,  "9  times  4  are  36  ;  9  times  1 
are  9  and  3  are  12  ;  9  times  5  are  45  and  1  are  46,"  but  declare  the  results 
only. 


MULTIPLICATION.  33 


§  8.    Problems. 

Declare  the  answers  to  each  as  quickly  as  possible  with- 
out reading  the  problem  aloud  and  before  explaining. 

If  required  to  explain,  avoid  repetitions  and  unneces- 
sary ivords. 

65.  At  13^  each  what  will        Ans-    156^    Since  x  basket  costs 

^  ^  ,      ,  .  0      -^      ,    .  13^,  12   baskets  will  cost  12   times 

12  baskets  cost?     Explain.  13^  0r  156^. 

66.  Is  the  following  a  cor- 
rect explanation  of  Ex.  65? 

Since    1    basket   costs   W,    12       10N°;    Because  the  denomination  of 

,        12  is  baskets  and  not  cents. 
baskets  will  cost  13  times  VLr,  t 

or  156£ 

67.  At  $3  per  barrel  what  will  19  barrels  of  flour  cost? 

68.  At  19  per  head  what  will  21  sheep  cost? 

69.  If  a  man  travels  5  miles  an  hour,  how  far  will  he 
travel  in  16  hours?     Explain. 

70.  A  train  runs  29  miles  an  hour ;  how  far  will  it  run 
in  8  hours? 

71.  If  a  man  earns  $6  per  week,  how  much  does  he  earn 
in  12  weeks? 

72.  How  much  will  4  acres  of  land  cost  at  1 57  an  acre? 

73.  If  34  men  can  do  a  piece  of  work  in  11  days,  how  long 
will  it  take  one  man  to  do  it?     Explain.  1 

74.  If  3  pipes  fill  a  cistern  in  19  hours,  how  long  will  it 
take  one  pipe  to  fill  it  ? 

75.  Henry  is  12  years  old,  and  his  father  is  5  times  as  old ; 
how  old  is  his  father  ? 

76.  If  a  ship  sails  9  miles  an  hour,  how  far  will  it  sai] 
in  9  hours  ? 

AM.    MENT.   AR. 3 


34  AMERICAN  MENTAL   ARITHMETIC. 

77.  A    farmer    bought   19        An».  2Zf.    If  l  yard  cost  Sf,  19 

yards  of  cloth  at  3/  a  yard,     ^ards  cost  19  *?"*■  "  57^;  if  X 

^  .  ^  dozen  eggs  sold  for  16^,  5  dozen  sold 

and  gave  in  exchange  5  dozen  f or  ^ .  if  the  cloth  cost  67,  and  the 
eggs  at  16/  a  dozen;  how  eggs  brought  80^,  the  farmer's  due 
much  was  due  him  ?   Explain.     was  the  difference,  or  23^. 

When  the  scholar  has  thoroughly  mastered  this  form,  he  should  be  required 
to  abbreviate  the  explanation.  Thus,  the  cloth  cost  57^ ;  the  eggs  brought 
SOf ;  therefore  23^  was  due  the  farmer. 

78.  A  man  bought  7  yards  of  cloth  at  16/  a  yard,  and  5 
yards  of  cloth  at  12/  a  yard;  what  was  the  entire  cost? 

79.  If  the  income  from  one  cow  is  1 16  per  year,  and  from 
one  sheep  $2,  what  is  the  income  from  6  cows  and  8  sheep? 

80.  I  paid  18/  each  for  12  chickens,  and  2/  each  for  13 
eggs  ;  what  was  the  entire  cost  ? 

81.  What  will  be  the  cost  of  5  pictures  at  19/  each,  and  7 
hooks  at  3/  each  ? 

82.  A  lady  bought  6  dozen  buttons  at  12/  per  dozen,  and  gave 
in  payment  one  dollar ;  how  much  change  should  she  receive  ? 

83.  A  man  bought  6  barrels  of  apples  at  $3  per  barrel, 
and  gave  in  exchange  12  sacks  of  flour  at  1 2  per  sack;  how 
much  was  due  him? 

84.  James  bought  8  marbles  for  5/  each,  6  pencils  for  10/ 
each,  and  a  book  for  25/ ;  he  gave  in  payment  150/ ;  how 
much  change  should  he  receive  ? 

85.  Harvey  bought  9  oranges  for  7/  each,  and  11  lemons 
for  5/  each ;  he  gave  in  exchange  9  pounds  of  butter  at  15/ 
per  pound ;  how  much  was  due  him  ? 

86.  John  traded  8  marbles  worth  8/  each,  for  a  jack-knife 
worth  50/  and  some  money;  how  much  money  should  he 
receive  ? 


MULTIPLICATION.  35 

87.  What  is   the   profit  in         Arts.    $30.    It  is  best  to  find  the 

buying  6  cows  at  $20  each,      Profit  on  °f '    Thf  ■  «"  Profit  on 
,     °  ,,.  ~  ~-  ,  0      owe  cow  is  $  5  ;  on  6  cows,  $30. 

and    selling    at    $25    eacnr 

88.  What  is  the  profit  in  buying  10  shares  of  stock  at  $  99 
each,  and  selling  them  at  $103  each? 

89.  What  is  received  from  the  sale  of  16  cows  at  $  21  each, 
if  $  1  each  is  paid  the  agent  for  selling  them  ? 

90.  What  is  received  from  the  sale  of  16  shares  of  stock 
at  $101  each,  if  $1  each  is  paid  the  broker  for  selling  them  ? 

91.  How  much  does  a  man  gain  by  buying  6  cows  at  $21 
each,  paying  an  agent  $1  each  for  purchasing,  and  selling 
them  at  $  27  each,  paying  an  agent  $  2  each  for  selling  them  ? 

92.  Does  a  man  gain  or  lose  and  how  much  by  buying  6 
shares  of  stock  at  $101  each,  paying  a  broker  $1  each  for 
purchasing,  and  selling  at  $103  each,  paying  a  broker  $2 
each  for  selling  them? 

93.  When  beef  is  b$  a  pound,  and  pork  6^  a  pound,  how 
much  more  will  17  pounds  of  beef  cost  than  14  pounds  of 
pork? 

94.  Which  costs  the  more,  the  keeping  of  16  horses  9 
weeks  at  $1  a  week  each,  or  the  keeping  of  12  cows  12 
weeks  at  50^  a  week  each  ?     How  much  more  ? 

95.  Two  persons  start  from  the  same  point  and  travel  in 
opposite  directions:  one  travels  5  miles  an  hour,  and  the 
other  7  miles  an  hour ;  how  far  apart  are  they  at  the  end  of 
13  hours  ? 

96.  How  far  apart  are  they  at  the  end  of  13  hours  if  they 
travel  in  the  same  direction  ? 

97.  Which  is  cheaper  and  how  much  per  dozen,  to  buy 
eggs  at  25^  a  dozen  or  at  3^  each  ? 


DIVISION. 


Division  is  indicated  by  the  sign  -i-. 

The  number  to  be  divided  is  the 
dividend. 

The  number  by  which  to  divide  is 
the  divisor. 

The  result  is  the  quotient. 

That  which  remains  when  the  divi- 
sion is  not  exact  is  the  remainder. 

The  number  of  times  one  number 
is  contained  in  another  may  be  found 
by  subtracting. 

Division  is  a  process  shorter  than 
subtracting  for  finding  how  many 
times  one  number  is  contained  in 
another. 


Illustration. 

20  -*■  7  =  2f 
read 
20  divided  by  7  =  2  and 
6  remainder. 

20,  dividend. 
7,  divisor. 
2,  quotient. 
6,  remainder. 
To  divide  9  by  4,  by  sub- 
tracting. 

9-4-4  calls  for  the  num- 
ber of  times  that  4  may  be 
subtracted  from  9. 

9-4-4  =  1; 
that  is, 
9-4-4  =  2^ 


Declare  the  quotients  of: 

1.  144,  96,  36,  60,  72,  48,  24,  84, 108, 132, 120,  divided  by  12. 

2.  88,  44,  77,  132,  99,  121,  66,  22,  55,  33, 110,  divided  by  11. 

3.  72,  108,  81,  54,  63,  18,  45,  90,  27,  108,  36,  divided  by    9. 

4.  64,  56,  96,  24,  32,  16,  72,  40,  80,  48,  divided  by  8.    By  4. 

5.  63,  14,  35,  56,  21,  70,  84,  28,  42,  35,  49,  70,  divided  by  7. 

6.  54,  36,  18,  72,  54,  12,  24,  48,  42,  60,  divided  by  6.    By  3. 

7.  25,  40,  30,  15,  50,  35,  55,  20,  60,  45,  10,  65,  divided  by  5. 

8.  10,  18,  6,  16,  22, 14,  20,  8, 12,  4,  24,  28,  32,  divided  by  2. 

Ex.  1.   12,  8,  3,  6,  6,  etc.    Do  not  say,  "  144  -*- 12  are  12." 
3d 


DIVISION.  37 

Declare  the  quotient  and  remainder  of : 

9.  119,  111,  113,  81,  117,  86,  118,  77,  116,  90,  87,  -f-12. 

10.  109,  71,  106,  80,  102,  79,  105,  60,  107,  59,  86,  +11. 

n.  89,  60,  76,  83,  52,  63,  50,  73,  62,  55,  80,  25,  -*-  9. 

12.  79,  70,  61,  62,  73,  71,  67,  60,  78,  68,  63,  65,  -f-  8. 

13.  69,  59,  62,  54,  58,  23,  67,  57,  64,  55,  60,  53,  +  7. 

14.  59,  50,  41,  55,  51,  27,  57,  47,  46,  53,  45,  56,  -*-  6. 
is.  49,  41,  44,  38,  36,  22,  46,  39,  48,  34,  42,  37,  -j-  5. 

16.  39,  30,  33,  37,  26,  22,  34,  23,  31,  25,  29,  35,  -*-  4. 

17.  29,  19,  17,  23,  11,  16,  8,  28,  20,  22,  10,  13,    +  3. 
Ex.  9.   9,  11 ;  9,  3  j  9,  5  ;  6,  9  ;  etc.     Do  not  say,  "  119  -=- 12  are  9  and  11 

remaining." 

Declare  the  quotient  and  remainder  of : 

18.  12  contained  in  119,  17,  111,  14,  113,  13,  117,  18,  118, 
15,  116,  16,  114,  19,  112,  28,  110,  24,  115,  20, 109,  25, 106,  26, 
102,  21,  105,  23, 107,  22, 101,  27, 108,  29, 104,  34, 100,  30, 103, 
38,  99,  35,  97,  37,  90,  32,  93,  33,  98,  31,  96,  36. 

19.  12  contained  in  91,  39,  94,  40,  95,  48,  92,  44,  89,  41, 
85,  47,  80,  43,  88,  46,  84,  42,  81,  45,  86,  49,  82,  87,  53,  83,  57, 
79,  52,  73,  55,  77,  51,  74,  58,  78,  50,  76,  56,  70,  54,  75,  59,  71, 
64,  69,  68,  66,  62,  63,  60,  67,  65,  61,  72. 

20.  11  contained  in  109,  65,  106,  11,  102,  12,  105,  17,  107, 
14,  101,  13,  108,  18,  104,  15,  100,  16,  103,  19,  99,  28,  97,  24, 
90,  20,  93,  25,  98,  26,  96,  21,  91,  23,  94,  22,  95,  27,  92,  29,  89, 
34,  85,  30,  80,  38,  88,  77,  35,  71,  37,  66,  32,  69,  33,  79,  31,  54. 

21.  9  contained  in  89,  10,  80,  12,  84,  14,  86,  18,  87,  16, 
79,  28,  77,  20,  78,  26,  70,  22,  71,  29,  66,  30,  67,  35,  65,  32,  62, 
31,  64,  39,  54,  48,  50,  41,  51,  43,  52,  42,  53,  49,  45,  57,  46,  55, 
47,  58,  44,  56,  40,  59,  36,  68,  33,  60,  37,  61,  38,  63,  34,  69,  27. 

Ex.  18.    9,  11 ;  1,  5 ;  9,  3 ;  1,  2 ;  etc. 


38 


AMERICAN   MENTAL  ARITHMETIC. 


Read  the  quotient : 

22. 

23. 

12)119076324 

11)207806035 

25. 

26. 

7)369246810 

6)121416181 

28. 

4)245678900 

31. 

3)902637052 

34. 


29. 

5)312760030 

32. 

7)510000900 


8)507603295473286023 


36. 


6)102003040507623086 


38. 


4)816540092367813362 


40. 


9)923634373532146874 


42. 


12)823476298345151718 


44. 


5)345678987654321235 


46. 


9)876678543345210038 


24. 

9)803706256 

27. 

3)920212233 

30. 

8)405001762 

33. 

12)634310271 

35. 

7)123456780924572568 

37. 

5)102305067052342125 

39. 

3)276300009823145911 

41. 

11)382900768419582123 

43. 

6)102030456783961048 

45. 

4)953872641219382116 

47. 

11)398476521830057621 


over." 


Ex.  22.  9,  9,  2,  3,  0,  2,  7.     Speak  no  words  except  the  quotient  figures. 
i  not  say,  "12  into  119,  9  times  and  11  over;  12  into  110,  9  times  and  2 


DIVISION.  39 

State  results  rapidly: 

48.  7  times  14  are  how  many  times  2?  13?  12? 

49.  9  times  11  are  how  many  times  6?  12?     8? 

50.  5  times  15  are  how  many  times  3  ?  16  ?     9  ? 

51.  4  times  17  are  how  many  times  2  ?  15  ?     8  ? 

52.  2  times  33  are  how  many  times  3  ?  11  ?  22  ? 

53.  9  times  10  are  how  many  times  5  ?  18  ?  15  ? 

54.  6  times  16  are  how  many  times  3?  32?     6?  12?  24? 

55.  8  times    9  are  how  many  times  2?  18?     4?     6?     8? 
12?  24? 

56.  5  times  12  are  how  many  times  2?  3?  4?  6?  10?  151 

State  results  rapidly : 

9x    5,  +15  are  how  many  times  12?  15?     4?     7? 
14  x    6,  -   7  are  how  many  times  11  ?     7  ?  18  ?     6  ? 


5? 

7? 

9? 

6? 

8? 

7? 

3? 

6? 

2? 

8? 

6? 

3? 

57. 
58. 
59. 
60. 


63. 
64. 
65. 


L4  x  6,  -  7  are  how  many  times  11  ?  7  ?  18  ?  6  ? 
7x12,  -  8  are  how  many  times  19?  13?  14?  15? 
6  x    8,  +  -4  are  how  many  times  13?     6?     9?     7? 

61.  10  x   4,+   2  are  how  many  times    6?     8?     9?  12? 

62.  8x  9,  +16  are  how  many  times  8?  12?  16?  9? 
14  x  4,  +  8  are  how  many  times  16?  8?  4?  32? 
18  x  3,  +  6  are  how  many  times  15?  12?  5?  10? 
17  x   5,  +  5  are  how  many  times  18?  10?  16?     9? 

State  results  rapidly : 

66.  How  many  times  21  are  14  x    6  ?     7x9?     3  x  28  ? 

67.  How  many  times  24  are    6x16?     8x6?     6x12? 

68.  How  many  times  13  are  39  x    2?39x    3?26x    2? 

69.  How  many  times  12  are  15  x   4?     6x16?  54  x    2? 

70.  How  many  times  11  are  33  x    3?22x    4?44x    2? 

71.  How  many  times    9  are  12x12?     6x12?     5x18? 

72.  How  many  times  14  are    7x8?     2x49?     7x10? 


40  AMERICAN  MENTAL  ARITHMETIC. 

Declare  the  results  rapidly : 

73.  24-12;     25  +  5;    26  +  13;     27  +  9;     28-1-7;  30  +  15 
32-16;  33  +  11;  34  +  17;  35  +  5;  36  +  18;  38  +  19. 

74.  39  +  13;    40  +  8;    42  +  14;    44+4;    45  +  15;  48  +  16 
49  +  7;  50  +  10;  51  +  3;  52  +  4;  54  +  3;  55  +  11. 

75.  56  +  4;  57  +  3;  60  +  4;  64  +  4;  65  +  5;  66  +  6;  68  +  4 
70  +  7;  72  +  18;  75  +  5;  76+4;  77  +  11. 

76.  78  +  6;     80  +  5;     81  +  9;     84  +  6;     85  +  5;  88  +  11 
90  +  18;  91  +  13;  95  +  19;  96  +  16;  98  +  7;  99  +  11. 

Declare  the  results  rapidly : 

77.  96  +  16,  12,  24,  3,  8,  32,  48,  4,  6;  98  +  2,  7,  49,  14. 

78.  99  +  11,  33,  9,  3;  94  +  2,  47;  93  +  3,  31;  92  +  2,  23,  46, 
4;  91  +  7,  13;  90  +  9,  5,  18,  3,  45,  30,  2,  15,  6. 

79.  88  +  11,  2,  22,  44,  4,  8;  87  +  3,  29;  86  +  2,  43. 

80.  85  +  5,  17;  84  +  7,  21,  12,  14,  4,  6,  42,  2,  3,  28;  81  +  9, 
27,3;  80  +  16,8,10,5,40,  20. 

81.  78  +  39,  2,  13,  6,  3;  76  +  38,  2,  19,  4;  74  +  37,  2;  72  + 
18,  12,  24,  6,  3,  4,  36,  2,  4,  8,  9. 

82.  70  +  35,  7,  5,  14,  2;  64  +  8,  4,  16,  32;  63  +  9,  3,  7,  21. 

83.  54  +  6,  3,  9,  27 ;  48  +  8,  4,  6,  2,  24,  3,  16. 

Name  sets  of  two  numbers  each,  whose  product  is: 

84.  99,  98,  96,  95,  94,  93,  92,  91,  90,  88,  87,  86. 

85.  85,  84,  82,  81,  80,  78,  77,  76,  75,  74,  72,  70. 

86.  69,  68,  66,  65,  64,  63,  62,  60,  58,  57,  56,  55. 

87.  54,  52,  51,  50,  49,  48,  46,  45,  44,  42,  40,  39. 

88.  38,  36,  35,  34,  33,  32,  30,  28,  27,  26,  25,  24. 

89.  22,  21,  20,  18,  16,  15,  14,  12,  10,  9,  8,  6,  4. 

Ex.  84.  33  and  3  ;  9  and  11.  7  and  14  ;  49  and  2.  6  and  16  ;  8  and  12  ; 
4  and  24 ;  3  and  32  ;  2  and  48;  etc. 


DIVISION. 


41 


§  9.    Precedence  of  Signs 

What  is  the  value  of  6  +  4  x  5  ? 

Mathematicians  have  agreed 
to  use  the  sign  'x'  before  the 
sign  '+.'  or  '-.' 

What  is  the  value  of  6  -  4  -  2  ? 

Mathematicians  have  agreed 
to  use  the  sign  '+'  before  the 
sign  '+'  or  '-.' 

What  is  the  value  of  24 -=-4 
x2? 

There  is  no  agreement  as  to 
which  sign  shall  be  used  first. 
It  is  best  to  avoid  such  ex- 
pressions. 

What  is  the  value  of  6  -  4  +  8  ? 

It  makes  no  difference  in 
what  order  the  signs  '+'  and 
'-'  are  used. 


If  the  signs  are  used  as  they 
occur,  the  answer  is  50 ;  if  the 
sign  '  x '  is  used  first,  26. 


If  the  signs  are  used  as  they 
occur,  the  answer  is  1 ;  if  the 
sign  '  -7- '  is  used  first,  the  answer 
is  4. 


If  the  signs  are  used  as  they 
occur,  the  answer  is  12  ;  if  the 
sign  '  x '  is  used  first,  the  answer 
is  3. 


If  the  signs  are  used  as  they 
occur,  the  answer  is  10  ;  if  the 
sign  '  +  '  is  used  first,  the  answer 
is  10. 


Find  the  value  of : 

90.  6  +  8-5-2-3  +  2.  98. 

91.  72-6-64-*-8-3.  99. 

92.  6x8  —  12  +  4.  loo. 

93.  96-5-16  +  72-24-8.  101. 

94.  99-11-81-5-9  +  25.  102. 

95.  7  +  8-5-4  +  9x2-12-4.  103. 

96.  9+16-5-8-18-5-3  +  2x5.  104. 

97.  25  +  10-f-5-27-5-3.  105. 


8  +  4-7  +  6-9-5-3. 
18  +  15-3  +  72-24. 
30  +  5^5-36-3-8. 
92 -j- 23  +  87 -s-3  +  49-7. 
98-J-7-42-6  +  18-9. 
33-3-10-5-2-8-5-4. 
64x2-5-32  +  88-5-8-11. 
96-f-32  +  84-*-12-70-*-7. 


Ex.  90.  9.     Say  6,  10,  7,  9. 


42 


AMERICAN  MENTAL  ARITHMETIC. 


§  10.    Parenthesis  or  Bar. 


To  indicate  that  several 
quantities  are  to  be  subjected 
to  the  same  operation,  they 
are  written  within  curved 
lines  or  brackets,  or  under  or 
over  a  straight  line. 

Commas  are  sometimes 
used  to  indicate  that  the  signs 
are  to  be  used  in  the  order  of 
their  occurrence. 


Illustration. 
(6  +  3)  x  5, 

or 

6  +  3  x  5, 

means, 

the  sum  is  to  be  multiplied  by  5. 

read, 
the  expression  6  plus  3  times  5. 

6,  +  8,  +  2,  x  7 
means, 
to  6,  add  8,  divide  by  2,  multiply 
by  7. 


Find  the  value  of : 

106.  [(9x8  +  9)  +  9  +  5]  +  2  +  8. 

107.  (6  +  8)-r-2  +  (5-3)x2. 

108.  (7x5-3x8)  x 8  +  4. 

109.  (8x12-18x4)-h(9x7-19x3). 

Ex.  106.     15. 


Find  the  value  of : 

no.   9,   x8,  +  6,  -=-13,  x5,  -4-2,   x4,  +4,  -8,  +9,   x5. 
ill.   90,  +9,  -nil,  x2,  +6,  -8,   x7,  +4,  -5,   x8,   +2, 
-7,  x8,  +1,  -7. 

112.  19,  x4,  +  5,  -i-9,  x5,  +  6,  -17,  xl8,  +7,  +3,  -;-J6. 

113.  18,    x5,   +  6,  h-8,    xl2,   -44,   +8,    +-12,    x9,    +7. 

114.  17,  x  4,  +5,  +8,  +6,  +3,  +2,  +5,  +14,  -10,  x8, 
+  9,  +8,  -1,  +7. 

Ex.  110.  7.     Say  9,  72,  78,  6,  30,  15,  60,  64,  8,  17,  85. 


DIVISION.  43 

Find  the  value  of : 

115.  6,  +7,  x5,  +  5,  +  2,  +18,  x4,  +4,  +5,  +3,  x6, 
+  2,  -11,  X8,  +1,  -i-8,  x7,  +8. 

lie.  98,  +7,  +7,  +8,  +7,  +12,  x29,  +3,  +12,  +30, 
h-12,  xll,  x3,   +11,   +12,   -5,   +1. 

117.  16,  +17,  -18,  x6,  +1,  h-13,  xl4,  +2,  -10,  +25, 
+  18,  +22,  -15,  xl8,  +9,  +9,  -9. 

lis.  45,  -15,  x3,  x5,  x2,  -6,  x4, -5,  xl2,  -16,  x8, 
-9,  x5,  +10,  x7,  x2,  +7. 

119.  2,  +19,  -16,  +15,  +18,  -13,  -12,  +9,  +8,  -17, 
+  25,  -13,  +18,  -6,  -9,  -5,  +4. 

120.  18,  +17,  -19,  -12,  +23,  +48,  -16,  -19,  +13, 
-11,  +12,  -9,  +16,  -8,  +17,  -18,  -19. 

121.  13,  +12,  -11,  +15,  -14,  +22,  -19,  -3,  +8,  +9, 
+  17,  +13,  -6,  -9,  -12,  -25,  +18. 

122.  7,  x7,  +1,  +5,  x8,  +5,  +17,  x6,  +2,  +16,  x49, 
*-7,  x6,  +6,  +18,  +6,  x8,  +4,  +23,  +6. 

123.  19,  +11,  +13,  +2,  +16,  +5,  +14,  +18,  +4,  +7, 
f  13,  +10,  +9,  +12,  +14,  +16,  +6,  +8,  +11. 

124.  245-18-13-15-16-12-11-10-9-8-7-6-5 
_4_3_2-l-ll_14-17-25-8. 

125.  19,  +8,-7,-6,  +13,  -14,  +18,  -17,  +16,  +12, 
-14,  -15,  +9,  -8,  +7,  -6,  -11. 

126.  4,  x5,  +1,  +7,  x5,  +2,  +6,  -3,  +4,  x6,  +2, 
+  8,  x4,  +4,  +2,  x8,  +1,  +9,  x6,  +2,  +7. 

127.  5,  xl2,  +3,  +7,  x8,  -5,  -3,  +8,  x6, +1, +7,  x6, 
+  2,  +11,  x4,  x4,  +7,  +8,  -11,  -^17. 

128.  6,  x7,  +9,  -3,  +8,  x6,  +4,  +5,  x3,  +1,  +5,  x6, 
+  6,  +18,  xlO,  +1,  +7,  x6,  +4,  +11. 

129.  8,  x5,  +2,  +7,  x6,  +4,  +10,  xl4,  +6,  +9,  -8,  +7, 
x2,  +4,  +2,  xll,  +11,  +12. 


44 


AMERICAN  MENTAL   ARITHMETIC. 


24 

48 


4  =  6. 

8  =  6. 


24-4 

12-2 


0, 


24  -  4  =  6. 
48  -  4  =  12. 


24 
12 


4  =  6. 
4  =  3. 


24-4 

24-8 


24 
24 


4  =  6. 
2  =  12. 


§  11.    Principles 

Multiplying  both  dividend 
and  divisor  by  the  same  number 
does  not  affect  the  quotient. 

Dividing  both  dividend  and 
divisor  by  the  same  number  does 
not  affect  the  quotient. 

Multiplying  the  dividend 
multiplies  the  quotient. 

Dividing  the  dividend  divides 
the  quotient. 

Multiplying  the  divisor  di- 
vides the  quotient. 

Dividing  the  divisor  multi- 
plies the  quotient. 

Division  is  expressed  in  four 
ways  : 

By  writing  the  dividend  above 
and  the  divisor  below  a  hori- 
zontal line. 

By  writing  the  sign  4-f-'  be- 
tween the  terms. 

By  writing  the  sign  ' : '  be- 
tween the  terms. 

By  writing  the  divisor  at  the 

left,   and   the   dividend   at  the 

right,  of  a  curved  line. 

The  first  method  was  originally  used.  Later,  to  get  both  terms  on  the 
same  horizontal  line  the  dividend  was  written,  then  the  line  ' — '  with  a  dot 
over  it  for  the  dividend  and  a  dot  below  for  the  divisor,  then  the  divisor. 
Later  the  line  was  omitted. 


Eight  divided  by  three  is  ex- 
pressed : 


,  fractional  method. 


8  —  3,  common  method. 


8:3,  ratio  method. 


3)8,  working  method. 


DIVISION.  45 

§  12.    Problems. 

Declare  the  answer  to  each  as  quickly  as  possible  without 
reading  the  problem  aloud  and  before  explaining. 

If  required  to  explain,  avoid  repetitions  and  unnecessary 
words. 

130.  If   14  apples  cost   28**, 

what  will  1  apple  cost  ?  Explain.        Ans'  ^  If  u  &™les  cost  28?> 

1  apple  will  cost  as  many  cents  as 
Ans.  2f.     If    14  apples  cost  28^,  1      14  fe  contained  times  in  28,  or  2?. 
apple  will  cost  T\  of  28^,  or  If. 

131.  What  is  the  cost  of  1  yard  of  cloth  when  16  yards 
cost  96^? 

132.  A  man  divided  $  200  among  20  persons ;  how  much 
did  each  receive  ? 

133.  If  18  yards  of  cloth  cost  1 54,  for  how  much  must  it 
be  sold  per  yard  to  gain  $  36  ? 

134.  A  farmer  gave  18  barrels  of  flour,  worth  1 4  a  barrel, 
for  12  yards  of  cloth;  how  much  was  the  cloth  a  yard? 

135.  If  24  hours  equal  360  degrees,  how  many  degrees 
equal  1  hour? 

136.  If  360  degrees  equal  24  hours,  how  many  hours 
equal  1  degree  ? 

137.  Eleven  cows  were  sold  for  $  220  ;  what  was  the  sell- 
ing price  of  each  ? 

138.  Eleven  shares  of  stock  were  sold  for  $  1111 ;  what 
was  the  selling  price  of  each  ? 

139.  Eleven  cows  were  sold  for  $231,  and  $1  per  cow  was 
paid  to  the  agent  for  selling  them ;  how  much  did  the  owner 
receive  for  each  cow? 

140.  If  9  tables  cost  1 108,  what  will  12  tables  cost? 


46  AMERICAN  MENTAL  ARITHMETIC. 

141.  At   3^  each,  how  many  pears         Ans.  13  pears.    Since  1 

can  be  bought  for  39^  ?  Pear  costs  ^  39^  wiU  buy 

TTr      ,  ,      . .  .  .         , .  .  as  many  pears  as  3  is  con- 

142.  Would    this   explanation    be     tained  times  in  39}  or  13 

correct ?     "If  one  pear  costs  3^,  39^  pears. 

will  buy  as  many  pears  as  3^  U  con-  Yes>  Because  Sf  is  con- 

tained  times  in  39^,  or  13  pears"  tained  times  in  39^. 

143.  Would    this    explanation   be 

correct?     «  Since  1  pear  costs  3£  39^     co^i^t^  in'sfc  " 
will  buy  as  many  pears  as  3^  is  con- 
tained  times  in  39." 

144.  Would  it  be  right  to  say,  "If  1  pear  costs  3^,  39^  will 
buy  as  many  pears  as  3  is  contained  times  in  39^  "  ? 

145.  At  4^  each,  how  many  lemons  can  I  buy  for  72^? 

146.  If  1  cow  costs  %  15,  how  many  can  be  bought  for  175  ? 

147.  At  $19  each,  how  many  sheep  can  be  bought  for  $95? 

148.  A  and  B  started  together  in  the  same  direction  from 
the  same  point,  A  at  the  rate  of  5  miles  an  hour,  and  B  at 
the  rate  of  3  miles  an  hour ;  in  how  many  hours  will  A  be 
14  miles  ahead  of  B? 

149.  Traveling  as  before,  B  has  20  miles  the  start ;  in  how 
many  hours  will  A  overtake  B  ? 

150.  A  and  B  started  at  the  same  time  from  the  same 
point  in  opposite  directions,  with  the  same  rate  as  before  ; 
how  far  apart  will  they  be  in  10  hours  ? 

151.  After  traveling  10  hours  they  turn  around  towards 
home.  Who  will  reach  home  first?  How  far  will  A  have 
traveled?     How  far  will  B  have  traveled? 

152.  If  16  oranges  are  worth  32  pears,  and  3  pears  are 
worth  6  apples,  and  apples  are  worth  2$  each,  how  many 
oranges  can  be  bought  for  40^? 


DIVISION.  47 

153.  If  6  quarts  of  berries  cost  18*,  what  will  12  quarts 
cost? 

Ans.  36^.     1  quart  will  cost  £  of  Ans.  36^.     12  quarts  are  2  times 

18^,  or  3^  ;    12   quarts  will  cost  12      6  quarts ;  12  quarts  will  cost  2  times 
times  3f,  or  36f.  18^,  or  36^. 

154.  If  12  pounds  of  cheese  cost  108*,  what  will  36  pounds 
cost? 

155.  How  many  pounds  of  butter,  at  15  cents  per  pound, 
must  be  given  for  18  pounds  of  sugar  at  5  cents  a  pound? 

156.  If  8  sheep  cost  $  80,  how  much  will  24  sheep  cost  ? 

157.  If  5  men  can  do  a  piece  of  work  in  20  days,  in  how 
many  days  can  4  men  do  it? 

158.  How  many  barrels  of  flour  can  be  bought  for  $40,  if 
5  barrels  cost  $  50  ? 

159.  How  long  will  it  take  Paul  to  earn  99  cents,  if  he 
earns  18  cents  in  2  weeks  ? 

160.  How  many  years  will  it  take  to  pay  a  debt  of  $  1080, 
if  $ 720  are  paid  in  6  years? 

161.  How  much  will  24  barrels  of  apples  cost,  if  6  barrels 
cost  $24? 

162.  At  24*  for  12  apples,  what  will  72  apples  cost  ? 

163.  At  18*  for  3  dozen  clothes-pins,  how  many  clothes- 
pins can  be  bought  for  30*  ? 

164.  If  19  apples  cost  57/,  what  will  14  apples  cost  ? 

165.  If  19  apples  cost  57*,  how  many  apples  can  be  bought 
for  51*  ? 

166.  If  17  books  cost  $153,  what  will  22  such  books  cost? 

167.  At  $185  for  5  cloaks,  what  will  7  cloaks  cost? 

168.  If  23  cows  sell  for  $  920,  at  the  same  rate  what  wil) 
30  cows  bring  ? 


FACTORING. 


A  number  may  exactly  contain  an- 
other; the  container  is  a  multiple; 
the  contained,  &  factor. 

A  number  may  have  other  factors 
besides  itself  and  one;  a  composite 
number. 

A  number  may  have  no  other 
factors  besides  itself  and  one  ;  a  prime 
number. 

Several  numbers  may  have  no 
common  factor  greater  than  one; 
numbers  prime  to  each  other. 

Each  of  several  numbers  may  be 
prime  to  each  of  the  others ;  numbers 
severally  prime. 

Name: 

1.  All  the  composite  numbers  from  1  to  100. 

2.  All  the  prime  numbers  from  1  to  100. 

3.  Two  composite  numbers  prime  to  each  other. 

4.  Three  numbers  prime  to  each  other. 

5.  Three  numbers  severally  prime. 

Define  : 

6.  A  multiple  of  a  number.     9.   A  prime  number. 

7.  &  factor  of  a  number.       10.    Numbers  prime  to  each  other. 

8.  A  composite  number.         11.  Numbers  severally  prime. 

48 


Illustration. 

12  contains  6,  2  times. 

12,  a  multiple  of  6. 
6,  a  factor  of  '12. 

12,  a  composite  number. 
Its  factors  are,  1,  2,  3,  4, 
6,  12. 

7,  a  prime  number.  It 
has  no  factors  except  7 
and  1. 

8,  12,  25,  are  prime  to 
each  other. 

8,  9,  25,  49,  are  severally- 
prime . 


FACTORING. 


49 


A  number  is  divisible  : 

By  2,  when  its  last  digit  is 
divisible  by  2. 

By  5,  when  the  number  de- 
noted by  its  last  digit  is  divisi- 
ble by  5. 

By  4,  when  the  number  de- 
noted by  its  last  two  digits  is 
divisible  by  4. 

By  8,  when  the  number  de- 
noted by  its  last  three  digits  is 
divisible  by  8. 

By  3,  when  the  sum  of  its 
digits  is  divisible  by  3. 

By  9,  when  the  sum  of  its 
digits  is  divisible  by  9. 

By  11,  when  the  difference 
between  the  sum  of  its  digits  in 
the  odd  places  and  the  sum  of 
its  digits  in  the  even  places  is 
divisible  by  11. 

By  the  product  of  any  num- 
ber of  its  factors  which  are  sev- 
erally prime  to  each  other. 


Illustration. 

3960  is  divisible  by  2,  because 
0  is  divisible  by  2. 

3960  is  divisible  by  5,  because 
0  is  divisible  by  5. 


3960  is  divisible  by  4,  because 
60  is  divisible  by  4. 


3960  is  divisible  by  8,  because 
960  is  divisible  by  8. 

3960  is  divisible  by  3,  because 
18,  the  sum  of  its  digits,  is  divisi- 
ble by  3. 

3960  is  divisible  by  9,  because 
18,  the  sum  of  its  digits,  is  divisi- 
ble by  9. 

3960  is  divisible  by  11,  because 
0,  the  difference  between  9  (the 
sum  of  its  digits  in  the  odd  places) 
and  9  (the  sum  of  its  digits  in  the 
even  places)  is  divisible  by  11. 

3960  is  divisible  by  the  product 
of  3  x  4  x  5  x  11,  or  660,  because 
3,  4,  5,  and  11  are  factors  of  3960, 
and  are  severally  prime. 


Which  of  the  numbers  2,  3,  4,  5,  8,  9,  11,  are  factors  of : 

12.  27720?  15.   48532?  18.    72754? 

13.  3960?  16.    9768?  19.    3675? 

14.  6732?  17.   19998?  20.   14175? 

Ex.  12.   2,  3,  4,  5,  8,  9,  11. 

AM.    MENT.    AS. 4 


50  AMERICAN  MENTAL  ARITHMETIC. 

Name  all  the  following  numbers  that  are  factors  of  360360 

21.  2,  \  \\  7,  8,  \  11,  13,  17,  19,  23. 

22.  2x3,2x4,2x5,2x7,2x8,2x9. 

23.  2x11,2x13,3x4,3x5,3x7x2. 
24.,  3  x  9,  3  x  11,  3  x  13,  4  x  5,  4  x  7  x  3. 

25.  4  x  9,  4  x  11,  4  x  13,  5  x  7,  5  x  8  x  3. 

26.  5  x  11,  5  x  13,  7  x  8,  7  x  9,  7  x  11  x  2. 

27.  8x9,8x11,8x13,9x11,9x13. 

28.  2  x  9,  3  x  8,  4  x  8,  5  x  9,  7  x  13,  6  x  7. 

29.  2x3x8,2x4x5,2x5x9,6x9. 

30.  2  x  7  x  9,  2  x  8  x  11,  3  x  4  x  5  x  8. 

31.  3  x  4  x  11,  3  x  5  x  9,  3  x  7  x  9,  11  x  12. 

32.  9x11x13,7x11x13,5x11x13. 

33.  3  x  8  x  11,  3  x  11  x  13,  4  x  5  x  11  x  2. 

34.  4x7x11,4x9x11,5x7x9x4. 

35.  2  x  3  x  4,  2  x  3  x  5,  2  x  3  x  7,  20  x  7. 

36.  2  x  3  x  9,  2  x  3  x  11,  2  x  3  x  13  x  5. 

37.  2  x  4  x  7,  2  x  5  x  7,  2  x  5  x  8,  12  x  15. 

38.  2x5x11,2x5x13,2x7x8x5. 

39.  2  x  7  x  11,  2  x  7  x  13,  2  x  8  x  9  x  7. 

40.  2  x  9  x  11,  2  x  9  x  13,  2  x  11  x  13. 

41.  3x4x7,3x4x8,3x4x9x7x11. 

42.  3x4x13,  3x5x7,3x5x8x11x2. 

43.  3  x  5  x  11,  3  x  5  x  13,  3  x  7  x  8  x  9. 

44.  3  x  7  x  11,  3  x  7  x  13,  3  x  8  x  9  x  2. 

45.  3  x  8  x  13,  3  x  9  x  11,  3  x  9  x  13  x  5. 

46.  3  x  5  x  11  x  13,  4  x  8  x  9  x  11  x  13. 

47.  5x7x8x13,5x7x8x11x9x13. 

48.  5  x  8  x  9  x  11,  5  x  8  x  9  x  13  x  2  x  3. 

49.  3x5x9x11,3x4x7x9x11x13. 

Ex.  29.   2x5x9,  because  2,  5,  9  are  severally  prime. 


FACTORING.  51 

Of  the  following,  some  of  the  factors  are  given.  Find 
two  or  three  more  for  each,  by  taking  the  product  of  factors 
severally  prime : 

50.  Number  360 ;  factors  4,  9,  8. 

51.  Number  1155 ;  factors  3,  7,  5,  11. 

52.  Number  1260  ;  factors  12,  15,  7. 

53.  Number  600  ;  factors  3,  4,  10,  5. 

54.  Number  210 ;  factors  15,  14,  7,  2. 

55.  Number  660 ;  factors  20,  33,  3. 

56.  Number  2520 ;  factors  5,  7,  8,  9,  4,  3,  6. 

57.  Number  2431 ;  factors  11,  13,  17. 

Ex.  53.     12  or  3  x  4,  30  or  3  x  10,  15  or  3  x  5,  etc. 

By  inspection  tell  why  : 

58.  1224  is  divisible  by  72.  63.  2034  is  divisible  by  18. 

59.  3465  is  divisible  by  55.  64.  1463  is  divisible  by  77. 

60.  2394  is  divisible  by  63.  65.  3144  is  divisible  by  24. 

61.  10208  is  divisible  by  88.  66.  980  is  divisible  by  35. 

62.  10197  is  divisible  by  99.  67.  1728  is  divisible  by  72. 

Ex.  58.  1224  is  divisible  by  8  and  by  9 ;  hence  by  8  x  9  because  8,  9  are 
severally  prime. 

By  inspection  determine  a  common  factor  of : 

68.  36,  48,  72.  73.  300,  250,  400. 

69.  77,  88,  121,  22.  74.   3260,  84,  96. 

70.  96,  56.  75.   395,  .95,  625. 

71.  360,  144,  9872.  76..  88,  84,  90. 

72.  235,  25.  77.   378,  117,  234. 


Illustration. 

4x6x8 

2 

4x6x8 
2 

4x6x8 

2 

4  x  6  x  16 

4  x  12  x  8 

8x6x8 

2)4x6x8 
2x6x8 

2)4x6x8 
4x3x8 

2)4x6x8 
4x6x4 

52  AMERICAN  MENTAL  ARITHMETIC. 

§  13.    Multiplication  and  Division. 

A  number  expressed  by  its 
factors  will  be  multiplied  by 
a  number,  if  any  one  of  its 
factors  is  multiplied  by  that 
number. 

A  number  expressed  by  its 
factors  will  be  divided  by  a 
number,  if  any  one  of  its 
factors  is  divided  by  that 
number. 

Multiply : 

78.  2x3x4  by  6.  83.  2x3x4  by  5  x 6. 

79.  7  x  5  x  8  by  7.  84.  3  x  4  x  5  by  2  x  3. 

80.  9  x  7  x  3  by  4.  85.  6  x  5  x  7  by  4  x  5. 

81.  8  x  2  x  5  by  3.  86.  8  x  3  x  9  by  2  x  6. 

82.  9x6x7  by  2.  87.  7  x  9x8  by  2x3. 

Ex.  78.   2  x  3  x  24,  2  x  18  x  4,  or  12  x  3  x  4. 

Ex.  83.  10  x  18  x  4,  or  2  x  15  x  24,  or  10  x  3  x  24,  etc. 

Divide : 

88.  9x18x6  by  3.  93.   54x64x18  by  18  x8. 

89.  12x10x8  by  5.  94.   72x96  x  48  by  24x8. 

90.  17x3x6  by  17.  95.   81  x 72x44  by  9x11. 

91.  14x18x12  by  6.  96.   76x24x34  by  19x17. 

92.  18  x  24  x  36  by  12.  97.  48  x  36x24  by  16x12. 
Ex.  88.  3  x  18  x  6,  or  9  x"  6  x  6,  or  9  x  18  x  2. 

Ex.  93.  3  X  8  x  18,  or  54  x  8  x  1. 


FACTORING. 

Divide : 

98. 

85x6x7  by  17.                     104.  64x7x9  by  56 

99. 

95x8x3  by  19.                     105.   70x2x6  by  14 

100. 

95x8x3  by  24.                     106.   90x3x7  by  54 

101. 

80x9x7x11  by  88.               107.    78x3x7  by  39 

102. 

96x12x7x11  by  112.           108.   75x4x8  by  50 

103. 

72  x  8  x  7  by  168.                    109.   52  x  8  x  7  by  91 

110.      9x12x14  by  36;  42;  108. 

ill.   56  x   8  x   9  by  64 ;  72 ;  168. 

112.   26  x    8x12  by  24;  32;  104. 

lis.   75  x  84  x  16  by  25 ;  16 ;  175. 

114.   96x35x17  by  48;  14;  672. 

lis.    84  x  20  x  16  by  28 ;  40 ;  480. 

116.   19  x  18x14  by  28;  36;  126. 

117.    75  x  42  x  28  by  56 ;  42 ;  100. 

lis.   30x70x20  by  14;  24;  210. 

119.   17  x  19x18  by  34;  38;  153. 

120.   20  x  21  x  22  by  28 ;  56 ;  154. 

121.   23  x  24x25  by  92;  30;  115. 

122.   26  x  27  x  28  by  13 ;  63 ;  117. 

123.    29  x  30x31  by  58;  62;  186. 

124.  32  x  33  x  34  by  88 ;  44 ;  136. 

125.   44x45x46  by  55;  22;  460. 

126.   47x48x49  by  47;  21;  112. 

127.   62  x  63x64  by  93;  16;  288. 

128.   Q5xQ6x  67  by  26 ;  67 ;  143. 

129.   68x69x70  by  69;  23;  115. 

53 


Ex.    98.  5x6x7.     85  + 17  =  5. 

Ex.  101.  10  x  9  x  7.    Factors  88  are  8  and  11 ;  80  -*-  8  =  10  ;  11  -r- 11  =  1. 

Ex.  127.  2  x  21  x  64.       Factors    93    are    31    and    3  ;     62  -r-  31  =  2  ; 
63  -j-  3  =  21. 


54 


AMERICAN  MENTAL  ARITHMETIC. 


§  14.     Greatest  Common  Divisor. 


Illustration. 


2 

72 

144 

108 

3 

6 

12 

9 

2 

4 

3 

12  x  3 


32)70(2 
64 
6 


25)75 
3 


G.  C.  D. 


G.  C.  D.  32,  70  is  2. 
G.  C.  D.  32,    6  is  2. 


G.  C.  D.  75,  36  is  3. 
G.  C.  D.    3,  36  is  3. 


The  G.  C.  D.  of  two  or  more 
numbers  is  the  product  of  all 
the  common  factors  which  may 
be  used  as  successive  divisors 
until  the  quotients  are  prime  to 
each  other. 

The  G.  C.  D.  of  two  numbers 
is  the  G.  CD.  of  the  smaller  and 
of  the  remainder  found  by  divid- 
ing the  greater  by  the  smaller. 

One  of  the  numbers  may  be 
divided  by  a  number  prime  to 
one  of  the  others  without  affect- 
ing the  G.  C.  D. 

By  the  second  principle,  find  the  G.  C.  D.  of : 

130.  64,  96.  134.  35,  75.  138.   46,  69. 

131.  56,  84.  135.  44,  90.  139. 

132.  72,  108.  136.  27,  84.  140. 

133.  24,  76.  137.  36,  75.  141. 

Ex.  130.  32.     The  G.  C.  D.  of  64  and  96  is  the  G.  C.  D.  of  64  and  32  (the 
remainder),  or  32. 

By  the  third  principle,  find  the  G.  C.  D.  of : 

142.  75,  96.  146.   36,  44.  150.   35,  91. 

143.  98,  72.  147.   22,  36.  i5i 

144.  46,  68.  148.    77,  91.  152 

145.  51,  72.  149.   80,  64 

Ex.  142.   3.     75  +  25  =  3. 
96,  or  3. 


40,  60. 
32,  48. 
38,  57. 


72,  56. 

33,  75. 

153.   45,  95. 

G.  C.  D.  of  75  and  96  is  the  G.  C.  D.  of  3  and 


FACTORING. 


55 


§  15.    Least  Common  Multiple. 


To  find  the  L.  C.  M.  of  two 
numbers,  divide  one  of  them  by 
their  G.  C.  D.,  and  multiply  the 
quotient  by  the  other. 

To  find  the  L.  C.  M.  of  more 
than  two  numbers,  find  the 
L.  C.  M.  of  two  of  them,  then  of 
the  result  and  a  third,  and  so 
on. 

If  one  of  the  numbers  exactly 
contains  another,  the  smaller 
may  be  neglected. 


Illustration. 
L.  C.  M.  10  and  12  is  60. 

12  -  2  =  6. 
10  x  6  =  60. 


L.  C.  M.  10,  12,  15  is  60. 
L.  C.  M.  10,  12  is  60. 
L.  C.  M.  60,  15  is  60. 


L.  C.  M.  12,  24  is  24. 

12  may  be  neglected. 


By  the  first  principle,  find  the  L.  C.  M.  of 


154.  12,14. 

155.  15,  12. 

156.  16,  20. 

157.  24,  32. 

158.  40,  50. 

159.  60,  80. 

160.  30,  26. 

161.  14,  21. 

162.  18,  20. 

163.  32,  36. 

164.  36,  40. 


165.  60,  72. 

166.  25,  30. 

167.  30,  40. 

168.  24,  27. 

169.  96,  84. 

170.  49,  63. 

171.  72,  96. 

172.  48,  52. 

173.  75,  80. 

174.  60,  72. 

175.  35,  42. 


176.  78,  52. 

177.  68,  85. 

178.  95,57. 

179.  90,  72. 

180.  80,  96. 

181.  42,  63. 

182.  84,  63. 

183.  56,  42. 

184.  76,  72. 

185.  50,  75. 

186.  65,  26. 


Ex.  154.    84.     The  G.  C.  D.  of  12,  14  is  2  ;  14  +-  2  =  7  ;  12  x  7  =  84. 


COMMON   FRACTIONS. 


§  16.    First  Conception  —  An  Expression  of  Division. 


Division  may  be  expressed 
by  writing  the  dividend 
above,  and  the  divisor  below, 
a  line.  Such  an  expression 
is  a  common  fraction ;  the  di- 
vidend is  the  numerator ;  the 
divisor,  the  denominator. 

The  numerator,  or  the  de- 
nominator, or  both,  may  con- 
tain fractions ;  such  an  expres- 
sion is  a  complex  fraction. 

We  sometimes  speak  of  a 
fraction  of  a  fraction ;  a  com- 
pound fraction. 

An  integer  plus  a  fraction 
is  a  mixed  number.     The  plus 
sign  is  usually  omitted. 
2. 

Analyze  -|.     5. 

Define  by  first  conception : 

1.  A  common  fraction. 

2.  A  complex  fraction. 

3.  A  compound  fraction. 

66 


Illustration. 

f ,  common  fraction. 

It  means  4-^-5. 

4,  numerator. 

5,  denominator. 
read,  4  -r-  5. 


4    f    t9i 


complex  fractions. 


f  of  f ,  a  compound  fraction. 


6f ,  a  mixed  number. 

f ;  3  is  the  numerator ;  4,  the  de- 
nominator ;  it  means  3-^-4. 

f  is  the  numerator ;  f ,  the  denom- 
inator ;  it  means  |  -t- 1 . 


4.  A  mixed  number. 

5.  The  numerator. 

6.  The  denominator. 


COMMON  FRACTIONS. 


57 


§   17.    Second    Conception  —  One    or    More    of    the 
Equal  Parts  of  a  Unit. 


A  unit  may  be  divided 
into  two  or  more  equal  parts, 
and  one  or  more  of  those 
parts  may  be  taken. 

The  number  denoting  into 
how  many  parts  the  unit  is 
divided  is  written  below  a 
horizontal  line,  and  is  the 
denominator. 

The  number  showing  how 
many  parts  are  taken  is 
written  above  the  line,  and 
is  the  numerator. 

The  whole  expression  is  a 
common  fraction. 

According  to  this  concep- 
tion, is  |  a  fraction?  No. 
It  is  called  an  improper  frac- 
tion, i.e.  not  properly  a  frac- 
tion. 

According  to  this  concep- 

2 

tion,  is  |  a  fraction  ? 
I 


Illustration. 

A  C       B 


AB  is  divided  into  8  equal  parts ; 
AG  contains  5  of  them;  AG  =  5 
eighths  of  AB;  expressed,  AG  —  f  of 
AB. 

f,  common  fraction. 
5,  numerator. 
8,  denominator. 
read,  5  eighths. 

It  means  that  a  unit  is  divided  into 
8  equal  parts  and  5  of  those  parts 
are  taken. 

},  read  one  half;  f,  read  3  quar- 
ters, or  3  fourths. 


It  is  impossible  to  divide  a  unit 
into  5  equal  parts  and  then  take  8  of 
them. 


No.     It  is  impossible  to  divide  a 
unit  into  £  equal  parts. 


Define  by  second  conception 

7.  A  common  fraction. 

8.  The  denominator. 


9.    The  numerator. 
10.    An  improper  fraction. 


58 


AMERICAN  MENTAL  ARITHMETIC. 


§  18.    Change  of  Form  —  To  Higher  Terms. 


Multiplying  both  numera- 
tor and  denominator  by  the 
same  number  does  not  change 
the  value  of  a  fraction. 

This  is  to  prepare  frac- 
tions for  addition  and  sub- 
traction. 


Illustration. 

A  B     C 

I    '    I    '   I    '    I 

AB  =  l  orf,  of  AC. 

1  =  1 

(multiplying  both  terms  by  2). 


Change  : 

11.  f  to  16ths. 

12.  I  to  12ths. 

13.  I  to  40ths. 

14.  T9g  to  48ths. 

15.  f  to  35ths. 


16.   4  to  36ths. 


to  24ths. 


17. 


12 


18.  ^3  to  39ths. 

19.  -fj  to  77ths. 

20.  T5¥  to  28ths. 


21. 


T9T  to  51sts. 


22.   11  to  56ths. 


23. 


if  to  60ths. 


24.    if  to  84ths. 


25. 


if  to  90ths. 


Ex.  11.    jf.     To  make  the  denominator  16,  we  must  multiply  it  by  4  ; 
multiplying  both  terms  by  4,  §  =  if. 


Reduce  to  equivalent  fractions  having  their  least  common 
denominator : 


26. 

1     3     5 
3'   4'  6* 

31. 

1    1    JL 

5'    7'   3  0" 

36. 

5        3       4 
l¥'  5  6'  y 

27. 

f '  i34'  28r 

32. 

1    _5_    _7 
¥'    12'   2¥* 

37. 

4        6        7_ 
33'  66'   11' 

28. 

g  >  1 1    jr. 

9'   18'  36* 

33. 

9'   3  6'  Y2- 

38. 

Y'   2  8'  T* 

29. 

163'   3  9'   52* 

34. 

4         7_       11 

2  5'   50'    10  0* 

39. 

16'  18'   8' 

30. 

1    1    1 

2'   3'  6* 

35. 

4    X    -5- 
Y'  Y7'    11* 

40. 

i\'  f'  eV 

Ex.  26.    TV,  T\,  if.     Multiplying  both  terms  of  §  by  4  ;  of  |,  by  3  ;  of  £, 
by  2. 


COMMON   FRACTIONS. 


59 


§  19.   Change  of  Form  —  To  Lower,  Terms. 
Dividing    both   numerator   and  Illustration. 


denominator  by  the  same  number 
does  not  change  the  value  of  a 
fraction. 

This  is  to  reduce  fractions  to 
their  simplest  forms. 

Which  fraction  is  the  more 
readily  comprehended,  ||-,  or  f  ? 
Why? 


J B 

p-n 


m 


AB  =  %,  or  I,  oiAC. 


(dividing  both  terms  by  2). 


smaller. 


Reduce  to  lowest  terms 

«•  »  A'flA'  A- 

42      _8_    10.    16     11    18 

**•     16'  20'   32'   33'   36* 

4.0       19     11     16     12     21 
*d*     ¥¥'  4?'  4  8"'  ¥¥'  5  0' 

44  14    18     2  9    11    18 

™     52'   5¥'   58'   68'  T2 ' 

45  11    15.    12     65    13 
*3,     22'   39'   65'  18*   91* 

*«•  ft  ft  ft'  il-  tt  • 

47. 

84"'   8  4'  "84'    8  4"'   8"4* 

12     32     64     48     72 
96'   96'    96'   96'   96* 

P    il    11     11      5 
T2'   96'    64'  F0'  ¥5* 


.30     41    11     30 
60'  60'    90'  f  5' 


48. 
49. 
50. 
51. 
52. 
53. 


24 


T8"'   98"'  3  0'   84' 


1£     4  1    14     13  2     _H_ 
39'    92'   96'    14¥'   121* 

112       14  4      1^5     _6_26_ 
336'    1T2^'   625'   1250* 


JUL    28.    11    _6_3_    XX    -31 
133'  98'   98'    112'  $4'  63* 

169     116.     2  21     2  51     2  8  9. 
13  0'    140'    150'    16  0'    llO* 

_46      _30_    _21_    JUL    JUL 
138"'   150'   200'    152'  253* 

_3  9_    _7_6        56,    1-0  5     _9_2_ 
351'   532'    385'    525'   368* 

»    ¥%'  iVV  S%%  A55'  AV 


54. 
55. 
56. 
57. 


59. 
60. 
61. 
62. 
63. 
64. 


.2  4.     3  61 


4  41     41 


6  0'   5  7  0^'    8  0  0'   8¥"0"'   8F0* 

5  21     5 11    121      8       21 
690'   720'    750'  ¥0'   30* 

6  4     12_5.     211     3  43.    512 
80'   500'    360'  ¥20'  ¥70* 

129     24     31     41     60     .71 
9  00'   3  0'  ¥2'  12'   6  6'   91* 

7  2     14.     93       5  6     77     __8_5_ 
18'   90'    102'   63'   84'   102* 

2T'   6¥'    l¥§"'   23166'   3¥3'  f ' 

65  J5JL      8 1_    4  21    111    12.1 
°3,     512'  T29'   850'   930'   BTS' 

66  2  21     777     101     245     _8_25_ 
DO*     669'   630'   161'   735'   1650* 


Ex.  41.   |,  f,  |,  \,  i.     Divide  both  terms  of  if  by  6 ;  of  /T,  by  9 ;  etc. 


60 


AMERICAN  MENTAL  ARITHMETIC, 


§    20.     Change   of    Form  — To   a   Whole    or    Mixed 

Number. 


A  fraction  is  an  expression  of  divi- 
sion. It  means  that  the  numerator 
is  to  be  divided  by  the  denominator. 

A  mixed  number  is  an  integer  plus 
a  fraction.  If  the  integer  is  re- 
duced to  an  equivalent  fraction  hav- 
ing the  denominator  of  the  fraction, 
the  two  parts  may  be  united. 

To  reduce  a  mixed  number  to  an 
improper  fraction  is  a  case  in  addi- 
tion of  fractions. 


Illustration. 

8   92 

£—3 

a    t». 


2f  =  2  +  f . 

2  =  |. 
+  t=f. 

therefore 
2§  =  f. 


Reduce  to  a  whole  or  mixed  number 


e?.  y»¥- 

"•  ft  4f 

75. 

5  5  68. 
13'    15' 

68.    ff,  f f . 

72.    fl,  |f. 

76. 

15  8.5_ 
16'  13' 

«»•    if.  ft- 

73.    |f  |f. 

77. 

M  81 
1^'    13" 

70-  ihii- 

74.    |f,  f|. 

78. 

.99  .9.6. 
1^'   15" 

Ex.  67.    13f     Performing 

the  indicated  division,  96  h 

r-  7  =  135. 

Reduce  to  an 

improper  fraction : 

79.    5,  4^. 

83.    5i|,  6^. 

87. 

Hi- 

so.   71  8|. 

84.   311  4  jy. 

88. 

m- 

81.   9f,  6|. 

85.    5i|,  4&. 

89. 

Hi- 

82.    8|,8f. 

86.    511   6if . 

90. 

5if. 

Ex.  79.    f ,  |f .    4  =  f  f ;  f  f  +  T5T  =  |f .    This  example  is  often  explained  i 
since  there  are  ||  in  1,  in  4,  there  are  4  times  \%t  or  f$- ;  f  f  +  T5T  =  Tf. 


COMMON  FRACTIONS. 


61 


§  21.    Addition  and  Subtraction. 

Before  fractions  can  be  added  or 
subtracted,  they  must  be  reduced  to 
equivalent  fractions  having  a  common 
denominator. 

The  least  common  denominator 
should  always  be  found. 


Illustration. 


A+A=tt- 


Find  the  value  of: 


91. 

2+3* 

98. 

A  4-    1 
6^9* 

105. 

6*  +  7f 

92. 

!+!• 

99. 

ll4_    A 
13^     6* 

106. 

8|+7f. 

93. 

t+fr 

100. 

tt+  f 

107. 

9f+8J. 

94. 

t+f 

101. 

8*+2f 

108. 

12f  +  8f. 

95. 

HI- 

102. 

2*+8* 

109. 

10|+9f. 

96. 

*+*• 

103. 

3i+4f 

110. 

18f+3f. 

97. 

♦+* 

104. 

5|+6f 

111. 

25f+6i 

:.  91 

• i+*=t+f«* 

Find  the  value  of  : 


112.  i—  J. 

113.  f-TV 

115. 
116. 

117-  !-tV 

118.  f-    |. 
Ex.  125.  31  - 


12' 
♦  "*■ 

2_     A 


119. 


2f 


126.    8J-2J. 


120. 

4|-lf 

127. 

n-2f 

121. 

8J-2i- 

128. 

8f-7|. 

122. 

H-H- 

129. 

Tf-6f. 

123. 

5|-3|. 

130. 

H-5f- 

124. 

2-l«. 

131. 

8J-6|. 

125. 

H'-  *• 

132. 

9f-4|- 

l  =  3xV-A  =  2if-T\  =  2H. 


62  AMERICAN  MENTAL  ARITHMETIC. 

§  22.   Multiplication  —  Universal  Case. 

Multiply     the     numerators  Illustration. 

for  a  new  numerator  and  the  A            D    C    B 

denominators  for  a  new   de-  I       '       '       '       ' 

nominator,      canceling     when  3  of  AB  =  AC. 

possible.  %  of  AC  =  AD  =  ±oi  AB. 

.-.  f  of  f  of  AB  =  I  of  AB. 
Mixed    numbers    should  2     3     1 

be     reduced     to     improper  £  x  J  ~  2' 

fractions.  .  2 

Find  the  value  of : 

133.  T3¥x^-;  ^xf;  &xf.  139.  f  of  18;  fx91;  |x65. 

134.  §x6;  8x|;  9xf.  140.  |  of  25;  f  of  24;  f  of  16. 

135.  fx-&;  7xf;  9x^.  141.  f  of  21;^  of  44;  TV  of  65. 
136. -J  x  16;  18xf;  72  x|.  142.  f  of  20 ;  ^  of  91 ;  ^  of  55. 

137.  84x^;  £x96;  |x48.       143.  fxf;  fxff;  |xTV 

138.  96xT\:  i|x84;  T9_X64.     144.  fof&j  £of£;  &ofl6. 

145.  T9eOfM;iiofi|;-lfof84. 

146.  3|x6f ;  4f  X5V;  51x12. 

147.  2f  X3J;  41x11;  6f  x^- 

148.  5lXll;  21x21;  3ix2|. 

149.  ljxli;  lfxli;  2|x2i 

150.  f  of  16+§  of  9  +  11x8. 

151.  1  Of  1  +  1  Off.  153.    I  X  If -f  Off. 

152.  f  Of  1-1  Of  1  154.    31Xf-fx^. 

Ex.  133.   -274.    Divide  both  16  and  14  by  2.    Speak  no  words  except  the 
result. 

Ex.152.  TV;  fof  *  =  tf ;  iof  i  =  i;  tf  -  *  =  A. 


COMMON  FRACTIONS.  63 

§  23.  Division  —  Universal  Case. 

Divide  the  numerators  for  a  new 
numerator  and  the  denominators  for  Illustration. 

a  new  denominator,  changing  to  equiva-  8      « =  4 

lent  fractions  ivith  their  least  common 
denominator,  if  necessary. 


f.+  l=A  +  A  = 


Mixed    numbers    should    be    re- 
duced to  improper  fractions. 


Find  the  value  of  : 

155.  6-^f.                  168.       J.+f  181.  5-S-3J. 

156.  18-^f.                  169.       £-s-f  182.  21-1-1. 

157.  27  -*-£.                 170.      $•*-£,  183.  21-61 

158.  84^-J-                171.       f  +  f.  184.  7|^81 

159.  63 -f                   172.       £-*■£.  185.  71  -h8f. 

160.  48 -r-^g.                173.     i6^--1/.  186.  9|-f-10f. 

161.  54-^V                ^4.       l^lf  187.  6f^81 

162.  77-f-^.                175.    21-3.  188.  9f-6f, 

163.  20 -|,                 176.    2f-=-4.  189.  4f-6i. 

164.  32-^.            177.      3-21.  190.  8f-f-6l 

165.  16 -T\.                178.       4^f.  191.  9f+3f 

166.  25-f-f                  179.    1|-1  192.  7f-t-2f 

167.  49 -f                  180.       5-31  193.  8§-r-41 

Ex.  155.   9.    6  -=-  §  =  y  -4- 1  =  9.     If  the   dividend  is  an  integer,  it  is 

easier  to  divide  the  numerator  and  multiply  the  quotient  by  the  denomi- 
nator.    Thus,  6  +  §  j  6-2  =  3;  3x3  =  9. 

Ex.186.    11.     9J-*-10|  =  Y- ¥  =  ¥■-*-¥  =  *!• 

Ex.188,   ff.     9* +  6*  =  ¥  +  ¥  =  **• 


64  AMERICAN  MENTAL  ARITHMETIC. 

§  24.    Problems. 

194.  At  \i  each,  what  will  18  ap-  Am.  12ft  If  1  apple 
pies  cost  ?  costs  ^  18  apples  will  cost 

195.  If  1  yard  costs  18*  what  will     18  tlmeS  *  °r  "* 

|  of  a  yard  cost  ?  ,  ^nf  f  •  H  ^ costs 

•  «r  18ft  J  of  a  yard  will  cost 

.4ns.  12ft     If  1  yard  costs  18ft  §  of  a  yard  i  0f  18ft  or  6^  ;  §  of  a  yard 

will  cost  §  times  18ft  or  12ft  will  cost  2  times  6ft  or  12ft 

196.  If  1  yard  COSts  18£  what  will  Ans.  99ft  If  1  yard  costs 
5|  yards  COSt  ?  18ft  5  yards  will  cost  90f ; 

4iw.  99ft    If  1  yard  costs  18ft  5 J  yards  will      *  of  a  ^ard  wiU  cost  ^  \ 
cost  5£  times  18ft  or  99ft  90^  +  9f  =  99ft 

The  right-hand  explanation  of  the  last  two  examples  is  objectionable 
because  it  explains  the  process  of  multiplication.  The  explanation  should 
simply  point  out  the  operation  to  be  employed. 

197.  At  %f  each,  what  will  16  oranges  cost  ? 

198.  If  1  yard  of  cloth  costs  25^,  what  will  f  of  a  yard  cost  ? 

199.  At  6^  a  yard,  what  will  6|  yards  of  cloth  cost  ?  What 
will  13 J  yards  cost  ? 

200.  At  6 \t  a  yard,  what  will  6  yards  of  cloth  cost  ? 

201.  If  a  man  earns  f  of  a  dollar  per  day,  how  much  will 
he  earn  in  20  days  ? 

202.  If  1  quart  pears  costs  14^,  what  will  ^  of  a  quart  cost  ? 

203.  If  1  apple  costs  |^,  what  will  f  of  an  apple  cost  ? 

204.  If  1  orange  costs  2j^,  what  will  12  oranges  cost? 

205.  At  \f  each,  what  will  20  pears  cost? 

206.  At  6§^  a  yard,  what  will  1\  yards  of  braid  cost? 

207.  If  1  shawl  costs  I121-,  what  will  15  shawls  cost? 

208.  At  5J^  each,  what  will  16  cocoanuts  cost  ? 

209.  If  a  boy  reads  b\  pages  of  a  book  an  hour,  how  much 
will  he  read  in  2 J  hours  ? 


COMMON   FRACTIONS.  65 

210.  At  \t  each,  how  many  apples  can  be  bought  for  18/  ? 

Aiis.  24  apples.     If  1  apple  costs  Ans.  24  apples.     If  1  apple  costs 

f  ?,  \%?  will  buy  as  many  apples  as  f      f  f,  1  cent  will  buy  f  apples  ;  18^  will 
is  contained  times  in  18,  or  24  apples.      buy  18  times  f  apples,  or  24  apples. 

211.  If  -|  of  a  yard  of  cloth  costs  18/,  what  will  1  yard  cost  ? 

Ans.  24?.     If  ^  of  a  yard  costs  18^,  If  5  yards  cost  15?,  what  will  1 

1  yard  will  cost  §  of  18^,  or  24/-.  yard  cost  ? 

Ans.  24 ?.     If  f  of  a  yard  costs  If  5  yards  cost  15?,  1  yard  will  cost 


\  I  of  a  yard  will  cost  £  of  18ft 


or  6? ;   f ,  or  one  yard,  will  cost  4      verting  5 ;  in  the  same  way  we  may 
times  Oft  or  24ft  invert  f  in  the  example  above. 

212.  At  J/  each,  how  many  apples  can  be  bought  for  20/? 

213.  If  |  of  an  apple  costs  2/,  what  will  1  apple  cost  ? 

214.  At  §/  each,  how  many  apples  can  be  bought  for  18/? 

215.  If  |  of  an  apple  costs  4/,  what  does  1  apple  cost  ? 

216.  If  1  apple  costs  §/,  how  many  apples  can  be  bought 
forf/? 

217.  If  |  of  an  apple  costs  |/,  what  does  1  apple  cost? 

218.  If  |  of  a  melon  cost  20/,  what  was  the  cost  of  the 
whole  melon? 

219.  If  |  of  a  house  cost  $  1200,  what  did  the  whole  house 
cost? 

220.  If  |  of  a  ship  is  worth  $  15,000,  what  is  the  value  of 
the  whole  ship  ? 

221.  If  |  of  a  farm  cost  $4200,  what  did  the  whole  farm 
cost? 

222.  If  |  of  a  store  is  valued  at  $6300,  at  what  is  the 
whole  store  valued? 

223.  If  ^   of  a  garden  cost  $120,  what  did  the  whole 
cost  ? 

AM.    MENT.    AR.  5 


66  AMERICAN  MENTAL   ARITHMETIC. 

224.  12  is  |  of  what  num*  Ans.  18.  If  12  is  §  of  some  num- 
ber ?  ber,  i  of  the  number  is  \  of  12,  or  6  ; 

Ans.  18.     If  12  is  §  of  some  num-  h  or  the  whole  number,  is  3  times 

ber,  that  number  is  12  +  §,  or  18.  °>  or  18' 


225.  I  of  20  is  f  of  what  number?  f  of  what? 

226.  I  of  40  is  I  of  what  number  ?  -^  of  what  ? 

227.  -J  of  24  is  -J  of  what  number  ?  |  of  what  ? 

228.  ^  of  52  is  I  of  what  number  ?  i|  of  what  ? 

229.  T8g  of  76  is  I  of  what  number  ?  |£  of  what  ? 

230.  T5T  of  85  is  ||  of  what  number  ?  f  of  what  ? 

231.  T36-  of  48  is  f  of  what  number?  ^  of  what? 

232.  I  of  45  is  f  of  what  number  ?  -2^  of  \7hat  ? 

233.  28t  of  84  is  if  of  what  number?  f  of  what? 

234.  I  of  56  is  I  of  what  number  ?  |-  of  what  ? 

235.  ^  °f  84  is  £  °f  what  number?  §  of  what? 

236.  T9g  of  80  is  T9g  of  what  number?  jf  of  what? 

237.  -|  of  50  is  £  of  what  number  ?     -|  of  what  ? 

238.  I  of  the  scholars  in  a  school,  or  30,  are  girls;  how 
many  are  boys  ? 

239.  A  boy  lost  J  of  his  kite-string,  and  gave  away  -J-  of 
the  remainder;  he  then  had  400  feet;  how  long  was  the 
string  at  first? 

240.  Joseph  is  12  years  old;  f  of  Joseph's  age  is  f  of 
John's  age  ;  how  old  is  John  ? 

241.  The  head  of  a  fish  is  15  inches  long;f|  of  the  length 
of  the  heac^  is  T3T  of  the  length  of  the  rest  of  the  body ;  what 
is  the  length  of  the  fish  ? 

242.  In  a  school,  J  of  the  students  study  algebra ;  ^  of  the 
remainder,  geometry  ;  and  the  rest,  or  12,  trigonometry ;  how 
many  scholars  are  there  ? 


COMMON   FRACTIONS.  67 

243.  At  3  for  5^,  how-many  apples  can  be  bought  for  20^? 

Ans.  12  apples.     If  3  apples  cost  .        '           ,          _.         _..          . 

„.    .,         .        ...        .    .  .  Ans.  12  apples.     Since  200  are  4 

50,   1  apple  will  cost  40 ;   as  many  orvrf      ...             .J 

,       ^  ,     ,        .* -  '            _  ,  times   50,   200  will  buy  4  times  3 

apples  can  be  bought  for  200  as  %  is  ,                       , 

.                   .    ~«        .J       .  apples,  or  12  apples, 
contained  times  in  20,  or  12  apples. 

244.  If  6  quarts  of  berries  cost  12^,  what  will  17  quarts 
cost? 

^4ws.  34^.     If  6  quarts  cost  120,  Ans.  340.     Since  17  quarts  are  V- 

1  quart  will  cost  \  of  120,  or  20  ;  17       times  6  quarts,  17  quarts  will  cost  -1/ 
quarts  will  cost  17  times  20,  or  340.         times  120,  or  340. 

245.  If  27  cans  of  tomatoes  cost  $  2.70,  what  will  9  cans  cost? 

246.  At  4  for  5^,  how  many  apples  can  be  bought  for  20^? 

247.  At  2  for  3^,  how  many  apples  can  be  bought  for  30^  ? 

248.  At  3  for  2^,  how  many  apples  can  be  bought  for  30^  ? 

249.  If  4  quarts  of  berries  cost  12J^,  what  will  6  quarts  cost? 

250.  If  6  quarts  of  berries  cost  12^,  what  will  18  quarts 
cost? 

251.  If  J  of  a  pound  of  prunes  costs  10^,  what  will  f  of  a 
pound  cost? 

252.  If  |  of  an  apple  costs  f  ^,  what  will  f  of  an  apple  cost  ? 

253.  If  5  cans  of  tomatoes  cost  60^,  what  will  13  cans  cost? 

254.  At  35^  a  dozen,  what  will  3  oranges  cost  ? 

255.  At  20^  a  dozen,  what  will  9  eggs  cost  ? 

256.  At  5^  a  score,  what  will  60  clothes-pins  cost  ? 

257.  At  $  36  a  dozen,  what  will  eight  pairs  of  shoes  cost  ? 

258.  If  16  men  can  earn  $  32,  how  much  can  75  men  earn  ? 

259.  If  |  of  a  ship's  cargo  is  worth  $  2400,  what  is  §  of  the 
cargo  worth? 

260.  If  15  sheets  of  paper  cost  10  cents,  what  will  24  sheets, 
or  one  quire,  cost? 


68 


AMERICAN  MENTAL   ARITHMETIC. 


261.  If  A  can  do  a  piece 
of  work  in  2  days,  and  B 
can  do  the  same  work  in  3 
days,  how  long  will  it  take 
them  working  together? 

262.  If  A  and  B  together 
can  do  a  piece  of  work  in  5 
days,  and  A  alone  in  8 
days,  in  how  many  days 
can  B  alone  do  the  work  ? 


Ans.  1  \  days.  In  1  day,  A  can  do  \ 
of  it ;  B  can  do  \  of  it ;  they  can  both 
do  the  sum  of  \  and  i,  or  £  of  it.  If 
they  can  do  £  in  1  day,  it  will  take  as 
many  days  to  do  f ,  or  the  whole,  as  $  is 
contained  times  in  |,  or  li  days. 

Ans.  13i  days.  In  1  day  both  can 
do  i  of  it ;  A  can  do  \,  and  Bi-|,  or  ¥% 
of  it ;  it  will  take  B  as  many  days  to  do 
the  whole  as  ^ 
££,  or  131  days. 


263.  One  pipe  will  fill  a  cistern  in  4  hours;  a  second 
pipe  will  fill  it  in  5  hours ;  how  long  will  it  take  both  io 
fill  it? 

264.  Two  pipes  together  fill  a  cistern  in  6  hours ;  the  first 
can  fill  it  in  10  hours ;  how  long  will  it  take  the  second  to 
fill  it? 

265.  Two  pipes  carry  water  into  a  tank,  and  a  third  carries 
water  from  it.  The  first  pipe  will  fill  it  in  2  hours,  the  second 
in  3  hours  ;  the  third  will  empty  the  tank  in  1 J  hours ;  if  the 
tank  is  empty  and  all  3  pipes  are  used,  in  what  time  will  the 
tank  be  full  ? 

266.  A  cistern  holding  70  gallons  has  a  pipe  by  which  15 
gallons  will  run  into  the  cistern  in  1  hour,  and  another  that 
will  discharge  10  gallons  an  hour;  when  both  are  running, 
what  part  of  the  cistern  will  be  filled  in  3  hours? 

267.  A  can  do  a  piece  of  work  in  3  days ;  B  can  do  the 
same  work  in  4  days ;  if  A  earns  $  2  a  day,  what  does  B 
earn  per  day? 

268.  John  is  16  years  old,  and  James  is  f  as  old  ;  how  old 
is  James? 


COMMON   FRACTIONS.  69 

269.  In  a  school  there  are  27  girls,  and  f  as  many  boys ; 
how  many  scholars  are  there  in  the  school  ? 

270.  A  boy,  having  80  marbles,  lost  -f  and  sold  T3B  of  them  ; 
how  many  had  he  left? 

271.  A  man  said  that  |  of  his  money  was  4  times  his 
week's  wages ;  he  had  $ 100 ;  what  were  his  week's  wages  ? 

272.  If  a  bushel  of  wheat  costs  $1-|,  and  a  bushel  of  corn 
$f,  what  is  the  difference  in  the  cost  of  5  bushels  of  each? 

273.  John  gave  away  |  of  his  marbles,  and  lost  f  of  the 
remainder ;  how  many  had  he  left  if  he  had  60  at  first? 

274.  In  traveling  72  miles  a  man  went  f  of  the  distance 
the  first  day,  J  of  the  distance  the  second  day,  and  the 
remainder  the  third  day;  how  far  did  he  travel  the  third 
day? 

275.  Of  a  flock  of  sheep  -J  are  in  one  field,  J  in  a  second, 
^  of  the  remainder,  or  14,  in  the  third ;  how  many  sheep  are 
there  in  the  flock? 

276.  After  spending  £  of  my  money,  and  losing  J  of  the 
remainder,  I  have  $  30  left ;  how  much  had  I  at  first  ? 

277.  By  selling  a  watch  at  a  loss  of  $  36,  I  lost  |-  of  its 
value  ;  what  was  its  val»e  ?  C  *  j 

278.  By  selling  a  watch  for  $36,  I  lost  £  of  its  value; 
what  was  its  value  ? 

279.  By  selling  a  watch  for  $  36,  I  gained  -J  of  its  value ; 
what  was  its  value  ? 

280.  A  man  bought  6  gallons  of  vinegar  at  12-|^  a  gallon, 
and  paid  for  it  in  oranges  at  36^  a  dozen  ;  how  many  oranges 
did  it  take  ? 

281.  Walter  is  -|  as  old  as  his  father,  and  f  as  old  as  his 
mother;  if  he  is  18  years  old,  how  old  are  his  father  and 
mother  ? 


70  AMERICAN   MENTAL   ARITHMETIC. 

282.  I  bought  100  pounds  of  sugar  at  4f  ^  a  pound,  and  paid 
for  it  with  codfish  at  12|/  a  pound;  how  many  pounds  of 
codfish  did  it  take  ? 

283.  How  many  barrels  of  flour,  at  $6-J  a  barrel,  must  be 
given  in  exchange  for  25  barrels  of  apples  at  $  3  a  barrel  ? 

284.  How  many  dozen  eggs,  at  12|^  a  dozen,  will  pay  for  5 
pounds  of  candy  at  10^  a  pound  ?  Qj 

285.  A  person  owning  |  of  a  ship  sold  I  ©T  his  share  for 
jgfiffi;  ??wliat  was  the  value  of  the  ship?  * 

286.  If  a  man  can  do  a  piece  of  work  in  12J  days,  work- 
ing 8  hours  per  day,  how  many  days  will  it  take,  working  10 
hours  a  day  ? 

287.  If  6  cakes  cost  15/,  what  will  7  cakes  cost? 

288.  If  |-  of  a  melon  costs  12^,  how  many  apples  at  2^  each 
will  buy  the  melon  ? 

289.  I  sold  a  cow  for  $  32,  which  was  |  of  her  cost ;  what 
was  the  cost? 

290.  A  man  sold  a  cow  at  a  loss  of  $16;  the  loss  was  |  of 
her  value  ;  what  was  her  value  ? 

291.  If  |  of  a  sum  of  money  is  T^-  of  the  value  of  a  horse, 
^Rind  \  of  the  value  of  the  horse  is  $  20,  what  is  the  sum  of 

money  ? 

292.  At  a  selling  price  of  $18  for  sheep,  \  the  cost  of  the 
sheep  was  lost ;  what  was  the  cost  ? 

293.  |  of  |  of  16  is  |  of  H  of  how  many  times  5  ? 

294.  A  lady  bought  10  pounds  of  raisins  at  12-^  a  pound, 
and  paid  for  them  with  currants  at  5^  a  pound ;  how  many 
pounds  of  currants  did  it  take  ? 

295.  If  8  pounds  of  soda  cost  9|^,  what  will  5  pounds  cost  ? 

296.  How  many  pounds  of  coffee,  at  33|^  a  pound,  must  be 
given  in  exchange  for  8  pounds  of  tea  at  66|^  a  pound  ? 


COMMON   FRACTIONS.  71 

297.  A  has  $1|,  and  B  $2|;  they  divide  what  they  both 
have  equally  between  two  persons;  how  much  does  each 
receive  ? 

298.  If  a  river  flows  2|  miles  in  3£  hours,  how  far  will  it 
flow  in  1  hour? 

299.  -|  is  |-  of  what  number  ? 

300.  |  is  ^  of  how  many  times  -fa  ? 

301.  |  of  3|-  is  fa  of  how  many  times  ^  ? 

302.  When  cheese  is  -fa  of  a  dollar  a  pound,  what  will  § 
of  a  pound  cost  ? 

303.  If  I  buy  turkeys  at  the  rate  of  5  for  $  3,  and  sell  at 
the  rate  of  8  for  $7,  how  much  will  I  gain  on  40  turkeys? 

304.  How  many  pigs  can  I  buy  for  $  75,  at  the  rate  of  3 
for  $7,  and  have  $5  left? 

305.  If  4  men  can  dig  a  ditch  in  16  days,  what  part  of  it 
will  three  men  dig  in  7  days  ? 

306.  If  a  man  were  twice  as  old,  \  of  his  age  would  be  20 
years;  how  old  is  he? 

^307.  B  gave  f  of  all  his  money  for  a  cow;  he  paid  $12 
for  hens,  which  was  f  of  all  the  money  he  had  left ;  how 
much  had  he  at  first  ? 

308.  I  bought  stock  at  $800 ;  f  of  this  is  f  of  f  of  2  times 
the  present  value  of  the  stock ;  what  is  its  present  value  ? 

309.  ^_  of  85\s  if  of  how  many  times ff  of  -|  of  50  T\ 

310.  -^  is  |  of  what  number? 

311.  hjfa  of  7T6^)is  |  of  what  number  ? 

312.  Frank  is  16  years  old ;  if  4  years  were  added  to  his  age, 
he  would  be  f  as  old  as  his  brother;  how  old  is  his  brother? 

313.  f  of  f  of  2  x  28  is  f  of  |  of  what  number? 

314.  |  the  sum  of  two  equal  numbers  is  20  ;  what  are  the 
numbers? 


DECIMALS. 


The  principles  of  the  decimal  nota- 
tion may  be  extended  to  certain  frac- 
tions by  placing  a  period,  called  a 
decimal  point,  after  units1  place,  and 
writing  the  fraction  to  the  right  as 
lower  orders  in  the  decimal  scale. 

The  orders  to  the  right  of  the 
decimal  point  are  tenths,  hundredths, 
thousandths,  ten-thousandths,  hundred- 
thousandths,  millionths,  etc.,  and  only 
those  fractions  which  have  these  de- 
nominators can  be  written  as  decimals. 

The  part  to  the  left  of  the  decimal 
point  is  an  integer;  the  part  to  the 
right,  a  decimal  fraction. 

The  fractional  part  is  not  read  on 
the  same  plan  as  the  integral  part. 

We  read  the  whole  as  an  integer 
for  the  numerator,  and  then  declare 
the  denomination  of  its  last  digit  for 
the  denominator. 

The  first  reading  represents  a  frac- 
tion plus  a  fraction ;  the  second,  their 
sum. 


Illustration. 

345.6789 

5,  in  units'  place. 

. ,  decimal  point. 


1  unit  =  10  tenths  (jg)  ; 
tenth  =  10    hundredths 


(TVk);    1  hundredth 


10 


thousandths  (T£§o)  5  etc* 


345.6789 

345,  integer. 

.6789,  decimal  fraction. 

.234567,  on  the  plan  of 
reading  integers,  would  be 
read  234  thousandths,  567 
millionths. 

It  is  actually  read,  234 
thousand  567  millionths. 


First. 


Second. 


rWff  +  Ttfro  oinr  —  iVo4o5oV< 


ITS' 


72 


DECIMALS. 


73 


To  write  a  fraction  whose  denomi- 
nator is  10,  100,  1000,  etc.  (that  is, 
a  decimal  fraction),  we  write  the 
numerator  in  the  usual  manner,  and 
indicate  the  denominator  by  the  aid 
of  a  decimal  point. 


.  3  _  -  .003 


CO 

D 

■• 

z 

CO 

< 

X 

CO 

CO 

H 

CO 

3 

Q 

Q 

X 

o 

z 

CO 

Z 

h 

I 

< 

^ 

I 

< 

z 

h 

IL 

o 

CO 

CO 

3 

o 

x 

CO 
Q 

CO 

h 

z 

2 

CO 

X 

h 

to 

X 

g 

h 

Q 

z 
< 

CO 

CO 

3 
O 

X 
h 

CO 

X 

CO 

I 
h 
z 

0 

o 

-1 

I 

CO 

X 

CO 

s 

u. 

z 

Q 

_i 

o 

z 

Q 

Q 

h 

Zi 

Q 

h 

z 

w 

0 

< 

UJ 

< 

(f) 

Ul 

< 

Ul 

z 

-1 

Ul 

z 

o 

_l 

cc 
Q 

z 

CO 

z 

CO 

3 
O 

cc 
Q 

z 

CO 

z 

£ 

5 
O 

I 
fc 

z 

cc 

a 

z 

CO 

3 

o 

z 

D 

z 

o 

-1 

I 
z 

cc 
o 

z 

o 

-1 

3 

Ul 

X 

3 

UJ 

z 

Ul 

3 

X 

Ul 

3 

Ul 

3 

1 

I 

1- 

h 

I 

H 

3 

•— 

h 

X 

H 

h 

X 

I 

h 

X 

CD 

2 

5 

6 

7 

8 

9 

3 

• 

6 

7 

8 

4 

5 

2 

1 

7 

8 

INTEGER. 

DECIMAL. 

Beginning  with  the  decimal  point,  numerate : 

1.  .23456789254.        4.  .365;  .4685;  .032;  .007. 

2.  .000240506782.        5.  .001;  .00024;  .007058. 

3.  .0000123045123.       6.  .20304;  .506;  .8075. 


Read: 

7.  Ex.  4. 

8.  Ex.  5. 

9.  Ex.  6. 


io.  .0000001.  13.  800.3065.  16.  49675.35. 
n.  .234|.  14.  40.087.  17.  3.000003. 
12.  60.005.    15.  2683.3.    18.  5.604002. 


Ex.  11.  234|  thousandths.  A  common  fraction  is  of  the  same  denomina- 
tion as  the  order  which  it  follows. 

Ex.  13.  800  and  3  thousand  65  ten-thousandths.  In  reading  mixed  num- 
bers, and  is  used  only  for  the  decimal  point. 


74 


AMERICAN  MENTAL  ARITHMETIC. 


§  25.    Reduction  —  Common  Fractions  to  Decimals. 

|  means  5  ~-  8  ;  performing  the 
indicated  operation,  we  obtain  .625. 

The  factors  of  10  are  5  and  2 ; 
hence,  if  a  fraction  has  any  factor 
other  than  2  and  5  in  its  denomina- 
tor, the  division  will  not  be  exact. 

In  such  cases,  the  usual  plan  is  to 

carry  out  the  division  a  few  places 

and  to  write  the  remainder,  or  to  write 

the  sign  «+'  instead  of  the  remainder. 

As  a  curiosity,  the  division  may  be  carried  out  until  the  quotient  begins 

to  repeat,  and  dots  may  be  placed  over  the  first  and  last  of  the  figures  which 

repeat,  as  in  2d  result.     The  part  which  repeats  is  then  called  a  repetend. 


19.   Reduce  J  to  a 

decimal. 

8)5.000 

.625 

20.   Reduce  f  to  a 

decimal. 

1st  result. 

7)2.000 

.285^,  01 

•  .285+ 

2d  result. 

7)2.000000 

.285714 

To  be  memorized : 
.50 


1 

2 

I  =  .20 
|  =  .831 


f=.40 


|  =  .75 
.80        J=.16f 
|  =  .621      j=.87j 


Give  the  decimal  equivalents  rapidly : 

a-  h h  h  h  h  %  h  h  h  h  23-  Vt>  I*  f>  h  h  t.  h  h  f  • 

421783211     | 
&   3'  P   &  6'  S>   5'  2'   5>   5* 


22.  |,  h  h  b  b  b  b  b  h  !•■  24 


etc.     Do  not  look  at  above  table. 


Reduce  to  decimals: 
25.    f  f,  f,  t,  f,  1,  I  £.  27.    £,  fa  £,  ft,  1J,  J3,  fl 

26-  I.  f  f.  ^>  ft>  A.  A.  A-    2«-  tV  iV  fe  A-  if  >  tV  A 

Ex.  25.  1st.    .14f ;  }  =  1  +  7  =.14J. 


DECIMALS.  75 

§26.    Reduction  —  Decimals  to  Common  Fkactions. 
Give  the  equivalents  in  common  fractions  rapidly : 

29.  .50;  .33 J  ;  .871.  33.  .871;  .66| ;  .40. 

30.  .12|;  .16};  .331  34.  .75;  .20;  .60. 

31.  .25;  .20;  .371.  35.  .88};  .62}j,87}. 

32.  .80;  .831;  .621  36.  .75;  .331  ;.66§. 

Ex.  29.   |,  I ,  |.      Having  memorized  the  table  in  the  previous  section, 
the  student  should  read  these  at  a  glance. 

Reduce  to  improper  fractions : 

37.  3.50;  2.75;  3.40.  45.  2.6;  5.8;  4.66|. 

38.  1.331;  3.66f ;  2.121  46.  3.621 ;  5.5  .  6.25. 

39.  3.371 ;  2.831;  1.16}.  47.  7.2;  8.75;  9.6. 

40.  4.25;  6.871;  9.2.  48.  5.66};  7.831;  6.871 

41.  7.50 ;  9.871 ;  4.621  49.  8.50 ;  5.621 ;  6.831 

42.  9.75;  5.831;  6.80.  50.  9.7;  8.621;  7.331. 

43.  3.16f ;  5.331 ;  8.80.  51.  6.66f ;  7.90;  5.16f. 

44.  3.121;  1.621;  9.30.  52.  8.80;  7.33J;  4.831 
Ex.39.  -V-.  3.37J-- 3|  =  V-. 

Reduce  to  common  fractions  : 

53.  .48;  .25;  .375.  57.  .58;  .265;  .735. 

54.  .960;.  .225;  .144.  58.  .875;  .3125;  .4375. 

55.  .625;  .200;  .16.  59.  .821;  .721;  .6|. 

56.  .516;  .750;  .90.  60.  .7J ;  .201;  .181 


76 


AMERICAN  MENTAL   ARITHMETIC. 


§  27. 


Per  Cent. 


When  the  denominator  is  100,  there 
are  three  ways  of  expressing  the 
fraction. 

1st.    As  a  common  fraction. 

2d.    As  a  decimal  fraction. 

3d.    By  the  per  cent  symbol. 

Per  cent  and  hundredths  are  inter- 
changeable. 

Change  to  %  : 

61.  .06;  .08;  .09. 

62.  500  hundredths. 

64.  .16;  .85;  .96. 

65.  4000  hundredths. 

66.  8365  hundredths. 
Ex.62.   500%.     Ex.66.    8365%. 

Give  the  equivalents  as  % 


Illustration. 
Six  hundredths  may  be 
written, 
T§7,  a  common  fraction, 
.06,  a  decimal  fraction, 
6  %,  read  6  per  cent. 
8%  =.08. 


Change  to  decimals : 

67.  5%;  8%;  23%. 

68.  233%  ;  4025%. 

69.  86%;  75%. 

70.  i%;  f%. 

71.  .000f  %  ;  2%. 

72.  200%  ;  65%. 
Ex.  68.   2.88:  40.25. 


73.  h  h  h  h  h  h  h  h  h  h  h  h  h  h  f 

74     1   1    JL    JL    i   i    J-    4   X    i   4 
'**     ¥'   6'    16'    12'  f'   6'    16'   12'   12'  ¥'   6* 

75.  11  2|,  3f ,  4|,  5f ,  6£,  8i,  9|. 

Ex.  73.    50  %,  33J  %,  etc.     See  p.  74. 

Give  the  equivalents  as  fractions : 

76.  871%,  66$%,  80%,  50%,  621%,  16|%,  20%,  60%. 

77.  16|%,  371%,  87-|-%,  831%,  40%,  75%,  25%,  331%. 

78.  1831%,  280%,  1371%,  125%,  2331%,  175%,  225%. 

79.  1161%,  220%,  140%,  375%,  4331%,  212£%,  416| %. 
Ex.  76.   1,  I,  f ,  etc.     See  p.  74. 


DECIMALS.  77 

§  28.     Short  Methods. 

To  multiply  when  the  sum  of  the  Illustration. 

fractional   parts  is   one,   and  the  in-  6I 

tegers  are  the  same. 


»i 


To    the    product    of   the    integer 

6x1 
and   th^   integer  increased   by  one,  6  x  a 

anr^x  the  product  of  the  fractions.  6x6 


79.    What  is  6f  x6^?  6  x  7  +  |  x  f  =  42ft. 

Find  the  value  of : 

so.       6Jx6J.  96.       ljxlf.  112.  8.3x8.7. 

81.  TJxTf  97.      9^x9J.  113.  9.2x9.8. 

82.  8Jx8J.  98.      6|x6f.  114.  7.2x7.8. 

83.  4£x4J.  99.      9Jx9f.  lis.  2.2x2.8. 

84.  6|x6f  ioo.       8lx8|.  116.  2.3x2.7. 

85.  5l*5i-  ioi.      5fx5|.  117.  85x85. 

86.  9fx9f  io2.       8fx8f  lis.  74x76. 

87.  3fx3i  103.  7-^x7^.  119.  83x87. 

88.  8|x8f  104.  8.5x8.5.  120.  92x98. 

89.  3^x3  J.  105.  7.4x7.6.  121.  27x23. 

90.  5-i- x  51  106.  9.3x9.7.  122.  38x32. 

91.  101x101  107.  4.8x4.2.  123.  54x56. 

92.  llixlli  108.  11.9x11.1.  124.  31x39. 

93.  991x991  109.  5.7x5.3.  125.  63x67. 

94.  10TVxl0ii  110.  4.6x4.4.  126.  58x52. 

95.  12^3x1211.  111.  10.9x10.1.  127.  4.75x4.25. 

Ex.  80.    42.25.  6  x  (6  +  1)  =  42  ;  .5  x  .5  =  .25. 

Ex.  117.   7225.  85  x  85  =  8.5  x  8.5  x  100  =  7225. 


16. 


78  AMERICAN  MENTAL   ARITHMETIC. 

To   multiply  or  divide   when  one 
number  can   be  readily  reduced  to  a  Illustration 

simple  fraction. 

Reduce  to  the  fraction  and  mul- 
tiply. 

Multiply : 

128.  24  by  .50.  134.  56  by  .871  140.  23  by  .331 

129.  36  by  .75.  135.  36  by  .831  141.  26  by  .66f . 

130.  84  by  .831  i36.  25  by  .60.  142.  27  by  .25. 

131.  72  by  .87-|.  137.  75  by  .331  143.  29  by  .50. 

132.  33  by  .331  i38.  21  by  .66f .  144.  30  by  .121. 

133.  24by.66f.  139.  48  by  .371  145.  31  by  .371 

Ex.  129.   27.     .75=  f;  36  x  f  =  27. 

What  is: 

146.  .50  of  16?  .871  of  40?  %50.  .40  of  32?  .60  of  13? 

147.  .331  of  60?  .621  of  64?  151.  .16|  of  25?  .331  0f  19? 

148.  .66 J  of  36  ?  .371  of  56  ?  152.  .20  of  53  ?  .25  of  35  ? 

149.  .25  of  52?  .121  of  72?  153.  .37}  of  22?  1.121  0f  16? 

154.  20%  of  50?   75%  of  16? 

155.  66f%  of  72?   25%  of  28? 

156.  16|%of42?  33^%  of  18? 

157.  40%  of  60?   371%  .of  16? 

158.  60%  of  20?  621%  of  24? 

159.  1831%  of  30?   116|%of36? 

160.  2121%  of  40?  325%  of  16? 

161.  16fxl8?  25x120?         163.   66fx48?   83^x28? 

162.  75x96?    371x64?  164.   75x40?   25x80? 

Ex.  159.   55.     183|%  =  y  J  V  of  30  =  55. 

Ex.  161.  300.     16f  x  18  =  .16f  x  IB  x  100  =  300. 


DECIMALS.  79 

What  is  the  cost  of : 

165.  120  yards  of  cloth  at  50^  a  yard?  33^?  25^? 

166.  12  yards  of  cloth  at  $1.16§  a  yard?  $2.75?  13.25? 

167.  25  yards  of  cloth  at  25^  a  yard?  12^?     87^? 

168.  72  dolls  at  25^  each?  33^?  37^?  50/?  62£tf? 

169.  72  chairs  at  83^  each?  87^?  20/?  40/?  60/? 

170.  24  lamps  at  $1.37*  a  dozen?  |2.75  a  dozen ? 

171.  60  chimneys  at  40/  a  dozen  ?  37^/  a  dozen  ? 

172.  9  vests  at  $  8.75  a  dozen  ?  $3.25  a  dozen  ? 

173.  24  spools  of  thread  at  $2.75  per  100?  $4.50  a  gross? 

174.  16  pounds  of  ham  at  8J/  per  pound?  12^/?  16§/? 

175.  3  dozen  lemons  at  6  for  12|^?  8  for  a  quarter? 

176.  48  lamp  chimneys  at  33^/  per  dozen  ? 

177.  40  tables  at  $1.25  each?  $1.75?  $1.66f  ?  $1.87£? 

178.  18  table-cloths  at  $9  a  dozen?  $12.50  a  dozen? 

179.  64  hammocks  at  $1.25  each?  $ .1.50?  $1,831?  $1.75? 

180.  28  oranges  at  25/^5Pdozen?  50/  per  dozen? 

181.  45  bananas  at  75/  per  dozen  ?  33^/  per  dozen  ? 

182.  15  plums  at  20/  per  dozen?  12  J/  per  dozen? 

183.  30  knives  at  60/  each?  75/  each?  80/  each? 

184.  20  bottles  of  ink  at  60/  per  dozen?  30/  per  dozen? 

185.  12  dozen  pens  at  16f/  per  dozen  ?  25/  per  dozen  ? 

186.  100  cloaks  at  $12J  each?  $50?  $75?  $20? 

187.  21  pair  of  shoes  at  $2.75  a  pair?  $3.25?  $5?  $2.50? 

188.  4  dozen  plates  at  $1.25  per  dozen?  at  121/  each? 

189.  8  dozen  cups  at  $4.37*  per  dozen?  $3.40  per  dozen? 

190.  60  marbles  at  10/  per  dozen  ?  25/  per  dozen  ? 

191.  80  chains  at  75/  each?  50/?  $2.25?  $3.66f  ?  $4.40? 

192.  50  glasses  at  25/  each?  331/ ?  871/?  371/ ? 

193.  75  baskets  at  12 J/  each?  16|/?  20/?  25/?  75/? 

Ex.  165.   $  60  ;  $  40  ;  etc. 


80 


AMERICAN   MENTAL   ARITHMETIC. 


Find  the  value  of: 


194.  24 -j- .66§. 

195.  24-66f%. 

196.  24 -66§. 

197.  210  -j- .50. 

Ex.  194.  3G 
Ex.  196.    A 


198.  210 -.331 

199.  210-66f%. 

200.  210 -=-75. 

201.  210-871 

Ex.195.   30. 


202.  210 -.25. 

203.  210  +  25%. 

204.  210-25. 

205.  210-831 


What  is : 

206.  21-60%?  By  371%?  214. 

207.  21 -=-50%  ?  By  75%  ?  215. 

208.  21 -=-871%?  By  16|%?  216. 

209.  20-66|%?  By  33l%?  217. 

210.  20  +  40%?  By  621%  ?  2i8. 

211.  20  +  831%?  By  50%?  219. 

212.  25  +  16f%?  By  331%?  220. 

221. 


213.    25-5-831%  ?  By  121%? 


450 

=  4900. 


33  +  150%?  By  1371%: 
42-140%?  By  175%? 
66-1831%?  By  75%? 
By  331? 

By  50%? 

By  66f  %  ? 

By  16f  %  ? 

By  121%? 


122£  +  25%? 


213  +  25% 


472 

230  +  2^7 


25% 


How  many  yards  can  be  bought: 

222.  For  $12  at  25^?  75^?  16f^?  66f^?  331^?  37^? 

223.  For  $12  at  60^?  87^?  831^?  12^?  50^?  &2±ft 

224.  For  $120  at  $1.25?  $1,331?  $1.66§?  $1.50? 

225.  For$120at$1.87^?  $3.75?  $2.50?  $1.60? 

226.  For$150at$1.20'?  $1.25?  $3,331?  $.75? 

227.  For  $150  at  $2.50?  $1,871?  $1.66f?  $1,331? 

228.  For  $150  at  $.121?  $3.16f?  $.331?  $1.25? 

229.  For  $140  at  $1.60?  $1.20?  $121?  |16|? 

230.  For  $140  at  $25?  $371?  $1,871?  $1.75? 
Ex.  224.  90  yards.     120  -*-  1.25  =  120  *.  f  =  90. 


DENOMINATE    NUMBERS  —  ENGLISH 
TABLES. 


MONEY. 

10  mills  (m.)  =  l  cent  (f) 
10  4  =1  dime  (d.) 

10  d.  =1  dollar  ($) 

110  =1  eagle  (E.) 

ENGLISH  MONEY. 
4  farthings  (far.  or  qr.)  =  1  penny  (t?.) 
12  d.  =1  shilling  O) 

20  s.  =1  pound  (<£) 

21  s.  =1  guinea  (G.) 

The  table  of  U.  S.  money,  like  the  tables  of  the  Metric  system,  has  the 
submultiples  :  Latin  mille,  1000,  mill ;  Latin  centum,  100,  cent ;  Latin 
decern,  10,  dime.  Ten  of  any  denomination  always  make  one  of  the  next 
higher. 

A  sovereign  =  £1. 

In  English  money,  the  symbols  £,  s.,  d.,  qr.,  are  the  initials  of  the  Latin 
equivalents,  libra,  solidus,  denarius,  and  quadrans. 

TROY  WEIGHT. 

24  grains  (gr.)  =  1  pennyweight  (pwt.) 

20  pwt.  =1  ounce  (oz.) 

12  Oz.  =1  pound  (lb.) 

AVOIRDUPOIS   WEIGHT. 

16  ounces  (oz.)         =1  pound  (lb.) 

100  lb.  =1  hundredweight  (cwt.) 

20  cwt.  =lton(T.) 

AM.    MENT.    AB. — 6  81 


STTfL 


mod 


>  APOTHECARIES     WEIGHT. 

3^XA36vu«i       20  grains  (gr.)  =  l  scruple  (3) 
(J       3  3  =  1  draft  (3) 

83  =1  ounce  (  3  ) 

12  5  =1  pound  (lb) 

■U*  l^  (^rn^S^W^TLvm  measurT^ 
60  minims  (^l)  =  1  fluid  drachm  (/  3) 
8/3  =1  fluid  ounce  (/J) 

16 /S  -       =  lpint(0.) 

8  0.  =1  gallon  (Cong.) 

Troy  weight  is  used  in  weighing  gold,  silver,  and  precious  stones  ;  avoirdu- 
pois weight,  in  weighing  nearly  everything  else ;  apothecaries'  weight,  in  retail- 
ing dry  medicines  ;  apothecaries'  fluid  measure,  in  retailing  liquid  medicines. 

1  long  ton =2240  lb. ;  1  carat =4  gr.  Carat  is  also  used  to  denote  the  num- 
ber of  parts  in  24  that  are  pure  gold.     E.g.  *  12  carats  fine'  means  \\  gold. 

Grain  is  from  grain  of  wheat ;  7000  gr.  =  1  lb.  avoirdupois  weight,  5760 
gr.  =  1  lb.  troy  or  apothecaries'  weight.  Pennyweight  was  the  weight  of  the 
English  penny  in  silver  ;  ounce,  Latin  uncia,  twelfth  ;  pound,  Latin  pe?ido, 
weigh  ;  scruple,  Latin  scrupulus,  a  little  stone  ;  dram ,  Greek,  drachma.  Cong. 
is  from  the  Latin  congius,  a  gallon  ;  O  is  the  abbreviation  for  Latin  octavus, 
one  eighth,  the  pint  being  one  eighth  of  a  gallon.  The  origin  of  symbols  of 
apothecaries'  weight  is  unknown. 

LONG  MEASURE. 

12  inches  (in.)  =1  foot  (ft.) 

3  ft.  =1  yard  (yd.) 

51  yd.  or  16*  ft.  =1  rod  (rd.) 

320  rd.  =1  mile  (mi.) 

surveyors'  long  measure. 
7.92  inches  (in.)  =1  link  (li.) 
100  li.  =1  chain  (ch.) 

66  ft.  =1  chain. 

80  ch.  =1  mile  (mi.) 


0  SUbo  0^  frtJrv  "  1   C*^W\ixX& 


DENOMINATE   NUMBERS  — ENGLISH   TABLES.  83 

Long  measure  is  used  in  measuring  lengths  and  distances.  For  short 
lengths,  the  foot  rule  or  yard  stick  is  commonly  used ;  for  long  distances, 
a  chain  4  rods  long.  The  surveyor'' s  chain  was  made  4  rods  in  order  that 
10  square  chains  might  equal  1  acre.  It  is  divided  for  convenience  into  100 
links  ;  4  rd.  =  66  ft.  =  792  in.  ;  hence,  7.92  in.  =  1  li.  The  engineer's  chain 
is  100  feet  long. 

Inch  is  from  Latin  uncia,  twelfth  ;  foot,  human  foot ;  yard,  a  twig ;  mile, 
Latin  mille  passuum,  1000  paces. 

A  line  =  j1^  in. ;  a  furlong  =  40  rd.  ;  a  fathom  =  6  ft. 

SQUARE  MEASURE. 
144  square  inches  (sq.  in.)  =  1  sq.  ft. 
9  sq.  ft.  =1  sq.  yd. 

30£  sq.  yd.  or  272$  sq.  ft.  =1  sq.  rd. 
160  sq.  rd.  =1  acre  (A.) 

surveyors'  square  measure. 
10000  sq.  li.   =lsq.  ch. 
10  sq.  ch.        =1A. 
640  A.  =lsq.  mi. 

36  sq.  mi.       =  1  township  (Tp.) 

Square  measure  is  used  in  stating  areas  anc  surfaces;  it  is  formed  by 
squaring  long  measure  ;  12  in.  =  1  ft.  ;  squaring,  144  sq.  in.  =  1  sq.  ft. ;  3  ft. 
=  1  yd.,  squaring,  9  sq.  ft.  =  1  sq.  yd.  ;  etc. 

A  perch  =  1  sq.  rd.  ;  a  rood  =  40  sq.  rd. 

CUBIC  MEASURE. 

1728  cubic  inches  (cu.  in.)  =  1  cu.  ft. 
27  cu.  ft.       .  =  lcu.  yd. 

WOOD  MEASURET  '  <->c/o 
16  cu.  ft.  =  1  cord  foot  (cd.  ft.) 
8  cd.  ft.   =1  cord  (cd.) 

Cubic  measure,  is  used  in  stating  volumes  or  contents;'  it  is  formed  by 
cubing  long  measure  ;  12  in.  =  1  ft.,  cubing,  1728  cu.  in.  =  1  cu.  ft. ;  3  ft. 
1  yd.,  cubing,  27  cu.  ft.  =  1  cu.  yd. 


)icn 


84 


AMERICAN  MENTAL   ARITHMETIC. 


CAPACITY. 


LIQUID  MEASURE 


DRY   MEASURE. 


2pt.  =1  qt. 


8  qt.  =1  peck  (pk.) 
4  pk.  =  l  bushel  (bu.) 


4  gills  (gi.)  =  lpint(pt.) 
2  pt.  =1  quart  ( 

4qtf         H  =lgalk)U  (gal'.) 

^Li^bi^Jiieasu^  i|  us$idi$(\^&asuring  liquids ;  dry  measure,  for  measuring 
dry  substances. 

Gill  is  from  Latin  gilla,  a  drinking- glass  ;  pint,  Spanish  pinta,  a  mark ; 
quart,  Latin  quartus,  a  fourth  ;  gallon,  derivation  unknown. 

For  convenience,  the  number  of  bushels  of  grain  and  seed  is  determined 
by  weighing  instead  of  by  using  a  bushel  measure.  Many  of  the  states  have 
fixed  by  statute  the  number  of  pounds  to  be  reckoned  a  bushel.     See  p.  86. 

The  barrel  does  jiot  contain  any  uniform  number  of  gallons ;  the  number 
of  gallons  is  usually"  written  on  OUB  Uf  lEsheads/  31£  gallons  is  the  standard.  \ 


TIME. 


60  seconds  (sec.)  =  l  minute  (min.) 


60  min. 

=  1  hour  (h.) 

24  h. 

=  1  day  (da.) 

7  da. 

=  1  week  (wk.)    N 

4  wk. 

=  1  month  (mo.) 

12  mo. 

=  1  year  (yr.)  ^ 

365  da. 

=  1  common  yr. 

366  da. 

=  1  leap  yr. 

100  yr. 

=  1  century  (cen.) 

Thirty  days  hath  September, 
April,  June,  and  November ; 
All  the  rest  have  thirty-one, 
Except  the  second  month  alone, 
Which  has  but  twenty-eight,  in  fine, 
Till  leap  year  gives  it  twenty-nine." 


DENOMINATE  NUMBERS  —  ENGLISH  TABLES.  85 

Every  year  exactly  divisible  bv  4,  cen\     1892>  a  leaP  year, 
temiial  years  excepted,  is  a  leap  year,    k    1893'  a  common  year' 

Ivery  centennial  year  exactly  divis-  \  2000,  a  leap  year, 
ible  by  400  is  a  leap  year ;  the  others  I  1900'  a  common  year, 
are  common  years.  / 

How  many  days  are  there  in  Mar.?  Nov.?  June?  Aug.? 
Sept.?   Feb.?   Apr.?   Jan.?   May?   July?   Oct.?  Dec? 

What  month  numerically  is  Aug.  ?  June  ?  Nov.  ?  Mar.  ? 
Dec?   Oct.?   July?   May?   Jan.?   Apr.?   Feb.?   Sept.? 

A  day  is  the  time  of  the  revolution  of  the  earth  upon  its  axis/^a  month  k 
the  time  of  the  revolution  of  the  moon  around  the  earthy  The  months  are ■. 
January  (31  days),  February  (28  or  29),  March  (81),  April  (30),  May  (31), 
June  (30),  July  (31),  August  (31),  September  (30),  October  (31),  November 
(30),  December  (31).  The  months  are  also  designated  by  the  ordinal 
numerals,  January  being  the  1st  month.  r-*v 
/in  business,  30  days  are  counted  as  a  month.) 

itryear  is  the  time  required  for  the  earth  toTevolve  around  the  sun.  It  is 
365  days  5  h.  48  min.  49.7  sec,  or  very  nearly  365 \  days.  Instead  of  reckon- 
ing this  part  of  a  day  each  year,  it  is  disregarded,  and  an  addition  is  made 
when  it  amounts  to  one  day,  which  occurs  about  every  fourth  year.  This 
addition  of  one  day  is  made  to  the  month  of  February.  Since  the  part  of  a 
day  that  is  disregarded  when  365  days  are  considered  as  a  year,  is  a  little 
less  than  one  quarter  of  a  day,  the  addition  of  one  day  every  fourth  year  is  a 
little  too  much,  and,  to  correct  this  excess,  addition  is  made  to  only  every 
fourth  centennial  year.  This  is  why  every  year  divisible  by  4  except  cen- 
tennial years  is  a  leap  year. 

Formerly  time  was  reckoned  very  inaccurately.  In  46  b.c.  Julius  Caesar 
reformed  the  calendar  and  made  the  year  consist  of  365J  days,  but  even  this 
was  not  absolutely  accurate,  and  in  1582  the  error  in  the  calendar  established 
by  him  (called  the  Julian  calendar)  had  increased  to  10  days ;  that  is,  too 
much  time  had  been  reckoned  as  a  year,  so  that  the  civil  year  was  10  days 
behind  the  solar  year.  To  correct  this  error,  Pope  Gregory  XIII.  ordered 
10  days  to  be  stricken  from  the  calendar.  The  day  after  the  3d  of  October, 
1582,  was  called  the  14th,  and  thereafter  only  those  centennial  years  which 
were  divisible  by  400  were  considered  as  leap  years. 


86  AMERICAN  MENTAL  ARITHMETIC. 

CIRCULAR   MEASURE. 

60  seconds  (")  =  1  minute  (') 
60'  =1  degree  (°) 

30°  =  lsign(S.) 

90°  =1  right  angle. 

12  S.,  or  360°    =1  circumference  (C.) 
Circular  measure  is  used  iu  measuring  angles.    -The  division  of  a  circum- 
ference into  360  parts  may  have  been  suggested  by  the  days  in  a  year. 

COUNTING.  PAPER. 

12  units  =1  dozen.  24  sheets   =1  quire. 

12  dozen  =  1  gross.  20  quires  =1  ream. 

12  gross  =1  great  gross.  480  sheets  =  1  ream. 
20  units  =  1  score. 

EQUIVALENTS. 

231  cu.  in.  =lgal. 

2150.4  cu.  in.  =1  bu. 

4  bu.  =5  cu.  ft.  (nearly). 

4  heaped  bu.  =5  struck  bu. 

7J  gal.  =1  cu.  ft.  (nearly). 

621  lb.  =wt.  1  cu.  ft.  water. 
(7  ft.)3  to  (8  ft.)3    =1  ton  hay. 
7  cu.  ft.  corn  in  ear =3  bu.  shelled  corn. 

24  h.  =360°. 

^  6$  K  =  1  teaspoonful. 

5760  gr.  =1  lb.  troy. 

*V>5  5760  gr.  =  1  lb.  apothecaries'. 

^  7000  gr.  =1  lb.  avoirdupois. 

**    In  most  states  : 

48  lb.  =  l  bu.  barley.  60  lb.  =  l  bu.  potatoes. 


56  lb.  =  l  bu.  corn.  56  lb.  =  l  bu.  rye. 

~*   O       32  lb.  =  l  bu.  oats.  60  lb.  =  l  bu.  wheat 


^ 


DENOMINATE  NUMBERS  — ENGLISH  TABLES.  87 

§  29.    Exercises  in  English  Tables. 

U.S.   MONEY. 

How  many :  How  many  .• 

1.  m.  make  $  1  ?  6.    m.  make  1  d.  ? 

2.  d.  make  |1?  7.    m.  make  IE.? 

3.  t  make  $1?  ^  8.   m.  make  1  il 

4.  i  make  1  E.  ?   t  0  9.   p  make  1  d.  ? 

5.  d.  make  IE.?  10.    $  make  IE.? 

ENGLISH  MONEY. 

How  many :  How  many  : 

11.  far.  make  X  1  ?  16.  d.  make  Is.? 

12.  s.  make  XI?  17.  d.  make  1  G.  ? 

13.  s.  make  1  G.  ?  18.  d.  make  XI? 

14.  far.  make  1  s. ?  19.  far.  make  Id.? 

15.  far.  make  1  #.  ?  20.  X  make  1  G.  ? 

TROY  AND  AVOIRDUPOIS   WEIGHTS. 
How  many :  How  many : 

21.  gr.  make  1  lb.  (troy)  ?  26.   gr.  make  1  lb.  (apoth.)  ? 

22.  gr.  make  1  oz.  (troy)  ?  27.    oz.  make  1  lb.  (apoth.)  ? 

23.  oz.  make  1  lb.  (troy)  ?  28.    cwt.  make  IT.? 

24.  pwt.  make  1  lb.  ?  29.   lb.  make  IT.? 

25.  pwt.  make  1  oz.  ?  30.   gr.  make  IT.? 


88 


AMERICAN   MENTAL   ARITHMETIC. 


apothecaries'  weight  and  apothecaries'  fluid 

MEASURE. 

How  many : 

36.  gr.  make  1ft? 

37.  gr.  make  1  5  ?&!/ 

38.  3  make  1ft? 

39.  3  make  1ft? 
35f  /3  make  1  Cong.  ?               40.  3  make  IS? 


LONG   AND    SURVEYORS'    LONG   MEASURE. 


How  many : 

41.  in.  make  1  yd.  ? 

42.  in.  make  1  rd.  ? 

43.  ft.  make  1  rd.  ? 

44.  ft.  make  1  mi.? 

45.  rd.  make  1  mi.  ? 


How  many :      / 

46.  li.  make  J  ch.  ? 

47.  li.  Vnake  1  rd.  ? 

48.  ch.  make  1  rd.  ? 

49.  ch.  maKKl  mi.? 

50.  in.  niake  In.? 


SQUARE   AND   SURVEYORS     SQUARE  MEASURE. 

How  many : 

56.  sq.  ch.  make  1  A.? 

57.  sq.  n\mal?e  1  sq.  ch.  ? 

58.  sq.  li.  nmke  1  sq.  rd.  ? 

59.  A.  maj^e  1  sq.  mi.  ? 

60.  sq.  ml.  make  1  Tp.? 


How  many : 

51.  sq.  in.  make  1  sq.  ft.  ? 

52.  sq.  in.  make  1  sq.  yd.  ? 

53.  sq.  ft.  make  1  sq.  rd.  ? 

54.  sq.  rd.  make  1  A.  ? 

55.  sq.  ft.  make  1  sq.  yd.? 


DENOMINATE  NUMBERS  —  ENGLISH  TABLES. 


89 


CUBIC   AND  aSOJOB^MEASURES. 


How  many : 

61.  cu.  in.  make  1  cu.  ft.? 

62.  cu.  in.  make  1  cu.  yd. 

63.  cu.  ft.  make  1  cu.  yd.  ?| 

64.  cu.  in.  make  1  cd.  ft.  ? 

65.  cu.  in.  make  3  cu.  ft.  ? 


How  many : 

66.  cu.  ft.  make  1  cd.  ? 

67.  cu.  ft.  make  1  cd.  ft.  ? 

68.  cu.  ft.  make  3  cu.  yd.  ? 

69.  cu.  ft.  make  2  cu.  yd.  ? 

70.  cu.  ft.  make  3  cd.  ? 


LIQUID    AND   DRY   MEASURES. 


How  many : 

71.  pt.  make  1  qt.  (dry)^ 

72.  qt.  make  1  pk.  ? 

73.  pt.  make  1  bu.  ? 

74.  pt.  make  1  pk.? 

75.  pk.  make  1  bu.  ? 


How  many : 

76.  pt.  make  1  qt.  (liquid)  ? 

77.  qt.  make  1  gal.  ? 

78.  qt.  make  1  bbl.  of  31 J  gal. 

79.  pt.  make  1  bbl.  of  45  gal.  ? 

80.  qt.  make  1  bbl.  of  40  gal.  ? 


MISCELLANEOUS. 


How  many : 

81.  da.  make  1  leap  year  ? 

82.  — etegrees-imke-4--Gr? 

83.  cu.  in.  make  1  gal.  ? 

84.  ^n^in^-make-4-ter? 

85.  v_degrees~m&ke  IS.? 


How  many : 

86.  units  make  1  dozen? 

87.  units  make  1  score  ? 

88.  sheets  make  1  quire?' 

89.  sheets  make  1  ream? 

90.  units  make  1  gross? 


ijw 


90  AMERICAN  MENTAL   ARITHMETIC. 

§  30.   Reduction  —  In  the  Same  Table. 
Reduce : 

91.  <£3  to  d.  96.  48  c?.  to  s. 

92.  14  s.  to  far.  97.  300  far.  to  s. 

93.  5  G.  to  s.  98.  3000  far.  to  £. 

94.  £2  3  s.  4 d.  to  far.  99.  f  *.  to  d. 

95.  3  G.  5  s.  6  d.  to  d.  loo.  Xf  to  d. 

Ex.  95.  822  d.    3  G.  =  63s. ;  68  s.  =  816  d.\  816  cf.  +  6  d.  =  822  d. 

Reduce : 

101.  2  lb.  3  oz.  2  pwt.  (troy)  to  pwt. 

102.  960  gr.  (troy)  to  lb. 

103.  -|  lb.  (apoth.)  to  lower  integers. 

104.  J  lb.  (avoir.)  to  lower  integers. 
105.   |  T.  (avoir.)  to  oz.  108.   1440  3  to  lb. 
106.-8  Cong,  to  O:                          -109.- 1  Congrtrr"t , 
107. ^Cong.  SO.  2/5  to/5-.  no.   3000  hi  lu/5, 

Ex.103.    43  63  134gr.;fib=4i3;  f  J  =  Gf37|3^1  ±-t>;  ft>=4gt. 


Reduce : 

111.  3  yd.  2  ft.  3  in.  to  in.  117.   |  rd.  to  lower  integers. 

112.  3  yd.  2  ft.  3  in.  to  ft.  118.    5  sq.  rd.  to  sq.  ft. 

113.  3  yd.  2  ft.  3  in.  to  yd.  119.    1  A.  to  sq.  yd. 

114.  0  etc  10  li.  lu  II.  120.    3500  cu.  in.  to  higher 

115.  160  cu.  ft.  to  cd.  integers. 

116.  1728  in.  to  rd.  121.    5184  cu.  in.  to  cu.  ft. 
Ex.  120.    2  cu.  ft.  44  cu.  in. 


DENOMINATE   NUMBERS  — ENGLISH   TABLES. 


91 


Reduce : 

122.  .6   bu.   to    lower    in- 
tegers. 

123.  96  qt.  to  bu. 

124.  2  bu.  2  pk.  1  qt.  to 
lower  integers. 

125.  252  gal.  to  pt. 

126.  1J  gal.  to  qt. 

Ex.  122.  2  pk.  3  qt.  .4  pt. ;  .6  bu. 

Reduce : 

131.  2  mo.  16  da.  to  da. 

132.  1  leap  yr.  2  da.  to  wk. 

133.  2500  min.   to   higher 
integers. 

134.  2  h.  2  min.  2  sec.  to 


127.  2000  pt.  to  higher  in- 
tegers. 

128.  |-   gal.   to    lower    in- 
tegers. 

129.  1728  pt.  to  higher  in- 
tegers. 

130.  1728  pt.  to  gal. 

:2.4  pk. ;  .4  pk.  =  3.2  qt. ;  .2  qt.  =  .4  pt. 


sec. 


135.   |   h.    to    lower    inte- 


136.  5  score  to  units. 

137.  6  score  to  dozen. 

138.  5  gross  to  dozen. 

139.  36  gross  to  great  gross. 

140.  960  sheets  to  reams. 

141.  1200  sheets  to  reams. 

142.  1200  quires  to  reams. 

143.  3°  4' 5"  to'. 


gers. 

Ex.  143.   184TV;  3°  =  180' ;  5"  =  Ty ;  180'  +f-+  TV  =  184TV 


Reduce : 

144.  £1  Is.  Id.  1  far.  to 
far. 

145.  |lb  to  lower  integers. 

146.  7  lb.  (troy)  to  lower 
integers. 

147.  1925  oz.   (avoir.)  to 
higher  integers. 

148.  480  ttl  to  3. 

Ex.  151.   T£o  mi. ;  33  ft.  =  2  rd 


149.  1  bu.  1  pk.  1  qt.  1  pt. 
to  pk. 

150.  63  gal.  to  bbl.  ' 

151.  33  ft.  to  a  fraction  of 
a  mi. 

152.  1  A.  to  sq.  yd. 

153.  1  cu.  yd.  to  a  fraction 
of  a  cd. 

=  3^  mi.  =  T^  mi 


92 


AMERICAN   MENTAL  ARITHMETIC. 


§  31.    Reduction  —  Table  to  Table. 


For  most  computations,  we 
may  consider  4  bu.  equal  to  5 
cu.  ft. 

For  most  computations,  we 
may  consider  1\  gal.  equal  to 
1  cu.  ft. 


4  bu.  =  2150.4  x  4  =  8601.6  cu.  in. 

5  cu.  ft.  =  1728  x  5  =  8640  cu.  in. 

7\ gal.  =  231  x7£ 
1  cu.  ft.  =  1728  cu.  in. 


Reduce : 

154.  20  cu.  ft.  to  bu. 

155.  20  bu.  to  cu.  ft. 

156.  400  bu.  to  cu.  ft. 

157.  400  cu.  ft.  to  bu. 

158.  32  bu.  to  cu.  ft. 

159.  100  cu.  ft.  to  heaped  bu. 

160.  80  heaped  bu.  to  cu.  ft. 

161.  64  heaped  bu.  to  cu.  ft. 

162.  75  cu.  ft.  to  heaped  bu. 

163.  8  ft.  x  8  ft.  x  3  ft.  to  bu. 

164.  7  ft.  x  10  ft.  x  2  ft.  to  bu. 

165.  4300.8  cu.  in.  to  bu. 

166.  6451.2  cu.  in.  to  bu. 

167.  4  «bu.  to  cu.  ft. 

168.  1  qt.  (dry)  to  cu.  in. 

169.  1  pt.  (dry)  to  cu.  in. 


170.  60  cu.  ft.  to  gal. 

171.  60  gal.  to  cu.  ft. 

172.  15  cu.  ft.  to  gal. 

173.  15  gal.  to  cu.  ft. 

174.  105  cu.  ft.  to  gal. 

175.  105  gal.  to  cu.  ft. 

176.  90  cu.  ft.  to  gal. 

177.  90  gal.  to  cu.  ft. 

178.  25  cu.  ft.  to  gal. 

179.  3  ft.  x  5  ft.  x  6  ft.  to  gal. 

180.  6  ft.  x  10  ft.  x  4  ft.  to  gal. 

181.  462  cu.  in.  to  gal. 

182.  1155  cu.  in.  to  gal. 

183.  4  gal.  to  cu.  in. 

184.  1  qt.  (liquid)  to  cu.  in, 

185.  1  pt.  (liquid)  to  cu.  in. 


Ex.  154.   16  bu. ;  1  cu.  ft.  =  $  bu. ;  20  cu.  ft.  =  20  x  f  bu.  =  16  bu. 
Ex.  155.   25  cu.  ft.  ;  1  bu.  =  f  cu.  ft. ;  20  bu.  =  20  x  f  cu.  ft.  =  25  cu.  ft. 
Ex.  160.    125  cu.  ft.  ;  1  h.  bu.  =  f  bu. ;  1  h.  bu.  =  ff  cu.  ft. ;  80  h.  bu. 
s  80  X  ft  cu.  ft.  =  125  cu.  ft. 


DENOMINATE  NUMBERS  —  ENGLISH   TABLES.  93 

Reduce  : 

186.  5000  gal.  water  to  cu.  ft.  191.  144  lb.  barley  to  bu. 

187.  10  cu.  ft.  water  to  lb.  192.  504  lb.  rye  to  bu. 

188.  125  lb.  water  to  cu.  ft.  193.  336  lb.  corn  to  bu. 

189.  6  cu.  ft.  water  to  lb.  194.  480  lb.  potatoes  to  bu. 

190.  720  lb.  wheat  to  bu.  195.  640  lb.  oats  to  bu. 

Reduce : 

196.  686  cu.  ft.  hay  to  T.  (73). 

197.  1024  cu.  ft.  hay  to  T.  (83). 

198.  3  T.  hay  to  cu.  ft.  (73). 

199.  3  T.  hay  to  cu.  ft.  (83). 

200.  21  cu.  ft.  corn  in  ear  to  bu.  shelled. 

201.  28  cu.  ft.  corn  in  ear  to  bu.  shelled. 

202.  50  cu.  ft.  corn  in  ear  to  bu.  shelled. 

203.  4  bu.  shelled  corn  to  cu.  ft.  corn  in  ear. 

204.  3  bu.  shelled  corn  to  cu.  ft.  corn  in  ear. 

205.  16  bu.  shelled  corn  to  cu.  ft.  corn  in  ear. 

206.  3  ft.  x  7  ft.  x  2  ft.  corn  in  ear  to  bu.  shelled. 

207.  40  cu.  ft.  corn  in  ear  to  bu.  shelled. 

208.  4  ft.  x  6  ft.  x  2  ft.  corn  in  ear  to  bu.  shelled. 

209.  75  cu.  ft.  corn  in  ear  to  bu.  shelled. 

210.  1.8  bu.  shelled  corn  to  cu.  ft.  corn  in  ear. 


Reduce : 

211. 

4  h.  to  degrees. 

217. 

14  h.  to  arc. 

212. 

4°  to  h. 

218. 

11  h.  to  arc. 

213. 

3°  to  time. 

219. 

6°  to  time. 

214. 

6  h.  to  arc. 

220. 

15°  to  time. 

215. 

60°  to  time. 

221. 

60°  to  time. 

216. 

360°  to  time. 

222. 

4  min.  to  arc 

DENOMINATE   NUMBERS --METRIC 
SYSTEM. 


The  Metric  system  is  a  decimal  system  in  which  all  the 
denominations  are  decimal  submultiples  or  multiples  of  a 
unit.  Ten  of  any  denomination  make  one  of  the  next  higher ; 
that  is,  the  multiple  is  10. 

The  submultiples  are  :  Latin,  mille  (1000),  centi  (100),  deci  (10).  The  mul- 
tiples are:  Greek,  Deka  (10),  Hecto  (100),  Kilo  (1000),  My  via  (10000);  the 
abbreviation  in  each  case  is  the  first  letter ;  small,  if  Latin  ;  capital,  if  Greek. 

^uomtdtiplesScOA/fi  Multiples.    v~\:Tl) 

10  milli  =  1  centi.  10  (units)  =  1  Deka. 

iti  sa  1  deci.       f"rt  |  a 


10  centi 
10  deci 


A, 


10  Deka 
10  Hekto 
10  Kilo 


1  Hekto. 
1  Kilo. 
1  Myria. 


Units  of  the  Different  Tables. 


Name. 
Long 


Land 


Area 


Weight 


Capacity 


Wood 


Unit. 


meter 


square 
meter 


gram 

liter 

stere 

94 


Abbre- 
viation, 


m. 


sq.  m. 


How  obtained. 


,0000001  distance 
equator  to  pole. 


10  m.  x  10  m. 


1  m.  x  1  m. 


wt.  1  cu.  cm. 
water. 


1  cu.  dm. 


1  cu.  m. 


Eng.  Eqniv. 


1.37  in. 


?V  acre 
nearly 


10  sq.  ft.  + 


15.432  gr. 


.908  qt.  dry 
1.05  qt.  liquid 


i  cord 
nearly 


1  sq.  Km.  = 
.3861  sq.  mi. 


1  Kg.  =  2\  lb. 


1  cu.  m.  water 
weighs  1  T. 


DElffdMIMTK/  NUMBERS  — 


METRIC    SYSTEM. 


95 


By  placing  the  various  units  in  the  table  of  submultiples, 
the  following  tables  are  formed.  Square  and  cubic  measure, 
as   in   English,   are   formed   by  squaring   and   cubing  long 


measure.         ru 
Long  Measure^ 

sb 

\o 

m 

& 

Capacity. 

Weight. 

10  mm.  =  1  cm. 

10  ml.  =  1  cl. 

"^10  mg.  =  1  eg. 

10  cm.   =  1  dm. 

10  cl.  =ldl. 

10  eg.   =^ldg. 

10  dm.  =  1  .m/CGbr 

10  dl.  =  l\hXuT\ 

IL.  10  dg.  =  1  g. 
^10  g.     =  1  Dg. 

10  m.     =  1  Dm. 

10  1.     =  1'D1. 

10  Dm.  =  1  Hm. 

10*  Dl.  =  1  HI. 

10Dg,  =  lHg. 

10  Hm.  =  1  Km. 

10  Hi.  -  1TC1. 

10  Hg.'=  1  Kg.^  * 

10  Km.  =  1  Mm. 

10  Kl.  =  1  Ml. 

,     10  Kg.  =  1  Mg. 

10  Mg.=  1  Quintal  (Q.). 
10  Q.    =1  Tonneau  (T.). 

Land. 

Square.       . 

Cubic.            G>  /C- 

10  ma.  =  1  ca. 

100 

sq.  mm.  =  1  sq.  cm. 

1000  cu.  mm.  =  1  cu.  cm.    i 

10  ca.  =  1  da. 

100 

sq.  cm.   —  1  sq.  dm. 

1000  cu.  cm.   =  1  cu.  dm. 

10  da.  =  1  a.,  etc.    100 

sq.  dm.  =  1  sq.  in. 

1000  cu.  dm.  =  1  cu.  m. 

Wood. 

100 

sq.  m.     =1  sq.  Dm. 

1000  cu.  m/    =  1  cu.  Dm. 

10  ms.  =  1  cs. 

100 

sq.  Dm.  =  1  sq.  Hm. 

1000  cu.  Dm.  =  1  cu.  Hm. 

10  cs.    =  1  ds. 

100 

sq.  Hm.  =  1  sq.  Km. 

1000  cu.  Hm.  =  1  cu.  Km. 

10  ds.   =  1  s.,  etc.    100 

sq.  Km.  =  1  sq.  Mm. 

1000  cu.  Km.  =  1  cu.  Mm. 

Land  measure  is  usually  given  100  ca.  =  1  a. ;  100  a.  =  l  Ha., 

—  the  other  denominations  being  omitted. 
Wood  measure  is  usually  given  10  ds.  =  l  s. 

The  measure  of  weight  has  two  extra  denominations :  the 
quintal  and  tonneau. 

The  following  denominations  in  italics  are  used  in  the 
metric  system  when  the  denominations  immediately  preced- 
ing them  would  be  used  in  the  English  system.  Foot,  yard, 
rod  —  meter  ;  mile  —  kilometer  ;  square  rod,  acre  —  are  ;  cord 

—  stere  ;  pound  —  kilogram  ;  pint,  quart,  gallon,  peck,  bushel 

—  liter  ;  ton  —  tonneau 


96 


AMERICAN  MENTAL   ARITHMETIC. 


= — OD 


00 


= — K 


CO 


The  metric  system  of  weights  and  measures  originated  in 
France  in  1795.  It  has  been  adopted  in  Austria,  Belgium, 
Brazil,  Denmark,  Germany,  Greece,  Holland, 
Sweden,  and  Switzerland.  Its  use  has  been 
authorized  by  the  Congress  of  the  United 
States  and  by  the  Parliament  of  Great  Britain. 
It  is  destined  to  displace  the  English  system, 
and  is  already  exclusively  used  in  scientific 
research. 

To  illustrate  long  measure,  prepare  a  strip 
of  paper  39|  in.  long ;  divide  it  into  10  equal 
parts  (dm.) ;  divide  each  dm.  into  10  equal 
parts  (cm.) ;  divide  the  first  cm.  into  10  equal 
parts  (mm.). 

To  illustrate  land  measure,  stake  out  on 
the  playground  a  square  10  m.  xlO  m.;  this 
square  will  mark  an  are. 

To  illustrate  wood  measure,  lay  off  on  the 
playground  a  square  1  m.xl  m. ;  at  each 
corner  drive  a  stake  leaving  1  m.  above 
ground;  these  stakes  will  hold  a  stere  of 
wood. 

To  illustrate  weights,  procure  a  piece  of  tin- 
foil that  weighs  as  much  as  a  nickel  5^  piece ; 
cut  the  foil  into  five  equal  parts ;  each  part 
will  weigh  1  g.  By  law,  the  nickel  must 
weigh  5  g.  Procure  a  stone  that  weighs  2 
lb.  3  oz. ;  it  will  weigh  a  kilogram  (nearly}. 

To  illustrate  measures  of  capacity,  draw 
on  paper,  rectangles  as  ABOB  (see  p.  97),  having  the 
dimensions   given  in  the  table;   complete  AEFD,  and  cut 


-Lfl 


=—  *? 


to 


=^-c\i 


DENOMINATE  NUMBERS  — METRIC   SYSTEM. 


97 


it  out;  roll  over  AD  to  BC,  and  paste  BUFC,  forming  cylin- 
ders ;  these  cylinders  will  hold  the  amounts  given  in  the  table. 


B     E 


Name. 

Length. 

Breadth. 

1  liter 

29. 16  cm. 

14.78  cm. 

5  deciliters 

23.15  cm. 

11.73  cm. 

2  deciliters 

17.10  cm. 

8.64  cm. 

1  deciliter 

13.54  cm. 

(5.80  cm. 

5  centiliters 

10.74  cm. 

5.44  cm. 

2  centiliters 

7.92  cm. 

4.01  cm. 

1  centiliter 

6.28  cm. 

3.18  cm. 

C    F 


1.  State  the  unit  of  long  measure,  how  it  was  obtained, 
and  its  English  equivalent. 

2.  Describe  in  full  the  unit  of  weight  (see  p.  94). 

3.  Describe  in  full  the  unit  of  land  measure. 

4.  Describe  in  full  the  unit  of  capacity. 

*""5.   Describe  in  full  the  unit  of  wood  measure. 
is    6.   Name  the  Latin  submultiples,  and  give  their  values. 

7.  Name  the  Greek  -s«bmultiples,  and  give  their  values. 

8.  What  is  the  multiple  in  all  the  measures  except  square 
and  cubic  measures  ?/  $What  is  the  multiple  in  square  measure  ? 

9.  What  is  the  multiple  in  cubic  measure  ? 

10.  Give  the  table  of  multiples  and  subnrultiples. 

11.  Give  the  table  of  long  measure.     Of  square  measure. 
Of  cubic  measure.     Of  capacity. 

12.  Give  the  table  of  land  measure  in  full.     Give  the  table 
of  land  measure  as  abbreviated.     Is  the  multiple  10  or  100? 

13.  Give  the  table  of  wood  measure  in  full. 

14.  Give  the  table  of  wood  measure  as  abbreviated. 

15.  Give  the   table  of  weight.     What  are  the   two  extra 
denominations  found  in  the  table  of  weight? 

AM.    MENT.   AR.  —  7 


98  AMERICAN  MENTAL  ARITHMETIC. 

§  32.    Practical  Questions. 

16.  How  tall  are  you  ?  How  long  is  your  arm  ?  What  is 
the  length  of  your  forefinger  ?  What  is  the  thickness  of  your 
thumb-nail  ?     What  is  the  width  of  your  thumb-nail  ? 

17.  What  is  the  width  of  this  room  ?  What  is  the  length 
of  this  room  ?  What  is  the  height  of  this  room  ?  What  is 
the  diameter  of  a  nickel  ?     What  is  the  thickness  of  a  nickel  ? 

18.  How  far  is  it  to  the  P.O.  ?  How  far  is  it  to  the  nearest 
city  ?  How  fast  does  a  passenger  train  run  ?  How  fast  can  a 
horse  pace  ?     How  fast  can  he  walk  ?     How  fast  can  he  trot  ? 

19.  What  is  your  weight?  How  much  beefsteak  would 
you  buy  for  breakfast  for  three?  What  is  the  weight  of  a 
nickel  ?  How  heavy  a  letter  of  the  first  class  will  go  for  2^  ? 
What  does  an  average  horse  weigh?  How  much  hay  will 
winter  a  cow?  How  much  coal  do  you  expect  to  burn  in 
your  kitchen  stove  this  winter  ? 

20.  How  much  milk  do  you  need  for  a  cup  of  coffee? 
How  much  milk  per  day  would  you  use  for  a  family  of  six  ? 
How  much  will  a  teacup  hold  ?  How  much  will  a  tablespoon 
hold?  What  is  a  good  yield  of  potatoes  to  the  acre?  What 
is  a  good  yield  for  wheat?     How  large  a  cistern  have  you? 

21.  How  much  land  do  you  need  for  a  flower  garden? 
What  is  the  area  of  this  floor  ?  How  much  land  is  needed 
for  a  good  farm  ? 

22.  What  is  the  contents  of  this  room  in  cu.  m.?  How 
much  earth  will  make  a  good  load  for  two  horses?  How 
many  cu.  m.  are  there  in  an  ordinary  rick  of  hay? 

23.  How  much  wood  will  an  ordinary  stove  burn  in  winter  ? 
How  much  wood  will  produce  as  much  heat  as  a  ton  of  coal  ? 

Note.  —  Give  answers  in  the  metric  system. 


DENOMINATE   NUMBERS  — METRIC    SYSTEM. 


99 


§  33.    Reduction — Metric. 

24.  Tell  whether  Latin  or  Greek,  and  give  meaning :  m, 
D.,  c,  M.,  K.,  d.,  H. 

25.  Give  the  Greek  for :  100,  1000,  10,  10000. 

26.  Give  the  Latin  for:  100,  1000,  10. 

27.  Read  102  mm. ;  468  cm. ;  3284  Kg. 

28.  Read  368  Kl. ;  5600  g. ;  123  s. ;  160  a. ;  238  Ha. 

29.  Read  397  Mg. ;  353  cu.  m. ;  189  sq.  mm. 

30.  Read  495  1. ;  672  Dl. ;  801  HI. ;  911  Kg. 

31.  Read  916  a. ;  210  cl. ;  717  dm. ;  111  Mg. 

32.  Read  495  Q.;  266  cu.  dm. ;  889  sq.  Dm. 


How  many : 

33.  mm.  make  1  Dm.  ?  45. 

34.  cm.  make  1  Mm.?  46. 

35.  dm.  make  1  Km.  ?  47. 

36.  m.  make  1  Hm.  ?  48. 

37.  mm.  make  1  Mm.?  49. 

38.  mm.  make  1  dm.  ?  50. 

39.  mm.  make  1  m.  ?  51. 

40.  mg.  make  1  Mg.  ?  52. 

41.  eg.  make  1  Hg.  ?  53. 

42.  Mg.  make  IT.?  54. 

43.  Kg.  make  1  T.?  55. 

44.  Dg.  make  1  Q.?  56. 


sq.  mm.  make  1  sq.  Mm.  ? 
sq.  cm.  make  1  sq.  Dm.  ? 
sq.  mm.  make  1  sq.  dm.? 
sq.  m.  make  1  sq.  Hm.  ? 
sq.  Dm.  make  1  sq.  Km.  ? 
cu.  m.  make  1  cu.  Mm.? 
cu.  mm.  make  1  cu.  Km.? 
cu.  Dm.  make  1  cu.  Mm.  ? 
dl.  make  1  KL? 
cl.  make  1  Ml.  ? 
HI.  make  1  ML? 
a.  make  1  Ha.? 


Ex.  33.   10000  ;  mille  =  1000  ;  Deka  =  10  ;  1000  x  10  =  10000. 

Ex.  38.       100  ;  mille  =  1000  ;     deci  =  10  ;  1000  --  10  =  100. 

Ex.  45.  1  with  14  ciphers  ;  mille  =  1000  ;  Myria  =  10000  ;  1000  x  10000 
='  10,000,000,  or  1  with  7  ciphers  ;  1  with  7  ciphers  squared  =  1  with  14 
ciphers. 


100 


AMERICAN   MENTAL  ARITHMETIC. 


§  34.   Reduction  —  English  and  Metric. 


Reduce  approximately : 

*  57.   16^Km.  to  miles.  /  b 

\   58.  $)  miles  to  Km.  J  £ 
^59.    3  m.  to  yards.  3  -^  ^ 
t^  60.    4  yards  to  m. 

^61.   600  m.  to  fee 
T^STMrnTtomTS  OQ7.  0  I 

L-  63.    320  in.  to  m. 

Ex.  57.    10  mi. ;  1  Km.  =  §  mi. 


Reduce  approximately : 

V  71.  5  gallons  to  1. 

\y  72.  30  1.  to  gallons. 

,/  73.  60  gallons  to  1. 

i/  74.  360  gallons  to  1. 

/  75.  126  1.  to  gallons. 

76.  1  bushel  to  1. 

77.  256  1.  to  bu. 

Ex.  71.   20  1.  ;  1  1.  =  1  qt. 


f  64.  600  feet  to  m.  — 

65.  2000  sq.  mi.  to  sq.  Km. 

66.  10,000  sq.  Km.  to  sq.  m. 

67.  16  acres  to  a. 

68.  400  a.  to  acres. 

69.  24  cords  to  s. 

70.  72  s.  to  cords. 

Ex.  64.   200  m. ;  1  m.  =  3  ft. ;  600  ft. 
=  200  m. 


78.  64  1.  to  bushels. 

79.  75  1.  to  quarts  (dry). 

80.  75  1.  to  quarts  (liquid). 

81.  10  1.  to  cu.  in. 

82.  25  1.  to  cu.  in. 

83.  40  1.  to  gallons. 

84.  100  1.  to  gallons. 


y* 


_    .  Reduce  approximately : 

85.  10  cu.  m.  of  water  to  tons. 

86.  50    cu.  cm.  of   water   to 
grains. 

87.  10  tons  of  water  to  cu.  m. 

88.  1    cu.  dm.  of    water    to 
pounds. 

89.  1  Kg.  to  pounds. 


Ex.  78.   2  bu. ;  1  1.  =  1  qt. 


90.  3  Hg.  to  pounds. 

91.  1  Hg.  to  ounces. 

92.  30  grains  to  g. 

93.  10  g.  to  grains. 
94. 
95. 
96. 


6  tons  of  water  to  cu.  m, 
5  bushels  to  1. 
18  Km.  to  miles. 


■I 


h 


PERCENTAGE. 


The  symbol  %  means  hundredths. 
Per  cent  and  hundredths  are  inter- 
changeable. 


Illustration. 

6%  =.06 

read 

per  cent  =  6  hundredths. 


§35. 

Change  to  %  : 

1.  .07,  .09,  .12. 

2.  .16,  .25,  .96. 

3.  325  hundredths. 

4*    TOO'  TOO'  TOO"* 

5.  9723  hundredths. 

6.  .001  .00f,  .0001  ^ 

Ex.6.    i%,  etc. 

Change  to  %  : 

13-    h  h  I'  i* 
"•   I'  h  h  t- 

1«5       4      1     £     1 

15*     5'  6'  6'  "8"' 

!«•  f't'4'f- 

17  2.     3    £     3 
-L/-  3'  ¥'   5'  ¥' 

18  5.    A     2     1 
•"*•  8'   6'   3'  T* 

Ex.13.   50%,  331%,  etc. 


Reduction. 

Change  to  hundredths : 

7.  17%,  13%,  14%. 

8.  70%,  90%,  77%. 

9.  466%,  100%,  325%. 

io.   \%  \%  \% 

11.  .0i%,  .001%,  .000J%. 

12.  200%,  4000%,  7283%. 
Ex.  11.    ,0|-  hundredths. 

Change  to  common  fractions 

19.  621%,  87J%,  37J%. 

20.  66f  %,  331%,  16}%. 

21.  621%,  25%,  40%. 

22.  75%,  60%,  25%. 

23.  20%,  80%,  331%. 

24.  2871%,  2331%,  116|%. 

Ex.  19.   &  |,  etc. 

101 


102  AMERICAN  MENTAL  ARITHMETIC. 

§  36.   The  Operation  Directly  Stated. 
What  is : 

25.  6  times  50?  42.   1331%  of  24? 

26.  |  of  50?  43.    266f%  of  18? 

27.  .06  of  50?  ^44.   325%  of  160? 

28.  6%  of  50?  45.   187!%  of  40? 

29.  121%  of  40?  46.   1121%  of  64? 

30.  331%  0f  60?  47.    331%  of  16? 

31.  16|%of36?  48.   2871%  of  16? 

32.  371%  of  16?  49.   125%  of  16? 

33.  831%  of  24?  50.   25%  of  34? 

34.  66f%  of  72?  jSi!   871%  of  17? 

35.  621%  of  32?  W   831%  of  25? 

36.  80%  of  20?  53.   66|%of49? 

37.  871%  of  48?  54.   1121%  of  33? 

38.  871%  of  40?  55.   2331%  of  12? 

39.  75%  of  144?  56.   16|%of36? 

40.  371%  of  64?  57.   121%  of  80? 

41.  25%  of  400?  58.    83J%  of  30? 

In  these  examples,  since  the  operation  is  directly  stated, 
no  explanation  is  required. 

Thus,  6  times  50  =  300  ;  f  of  50  =  30  ;  .00  of  50  =  3  ;  6%  of  50  =  3  ;  etc. 
To  explain  the  26th :  I  of  50  =  10,  £  of  50  =  3  x  10  =  30,  is  unnecessary, 
because  we  have  learned  how  to  multiply  50  by  f  in  a  previous  case,  and  we 
should  not  at  this  place  explain  that  process. 

Ex.  31.   10|  %  of  36  is  £  of  36,  or  6. 

Ex.  48.  287$  %  of  16  is  -2¥3  of  16,  or  46. 


PERCENTAGE.  103 

59.  If  a  rope  200  ft.  long 

_     .    .        r    .      F.  .      6  ^Lws.  190  ft.     It  shrinks  5%  of  200 

shrinks  5%   when  wet,  how     ft.,  or  io  ft. ;  200  ft.  -  10  ft.  =  190  ft. 
long  is  it  when  wet? 

60.  A  shepherd  having  240  sheep,  lost  16 1%  of  them  in  a 
storm  ;  how  many  had  he  left  ? 

61.  A  had  8200  and  gave  40%  of  his  money  to  B;  how 
much  did  A  retain  ? 

62.  A  mine  produces  2000  tons  of  ore;  25%  of  the  ore  is 
metal;  2%  of  the  metal  is  silver;  how  many  pounds  of  silver 
does  it  produce  ? 

63.  From  a  hogshead  containing  480  lb.  of  sugar,  66  J  % 
was  sold  at  one  time  ;  50%  of  the  remainder,  at  another  time ; 
how  many  pounds  remained  ? 

64.  A  dry  article  weighing  60  lb.  gains  10%  in  weight 
when  soaked  in  water ;  how  much  water  does  it  absorb  ? 

65.  The  population  of  a  town  in  1890  was  16J%  more  than 
1500 ;  what  was  the  population  ? 

66.  If  gunpowder  contains  75%  of  saltpetre,  10%  of 
sulphur,  and  15%  of  charcoal,  how  much  of  each  is  there  in 
40  lb.  of  powder? 

67.  Of  a  regiment  of  1000  men,  2%  are  killed,  7%  are 
prisoners ;  how  many  are  left  in  the  regiment  ? 

68.  Of  70  children  in  a  school  14^%  are  boys ;  how  many 
girls  are  there  in  the  school  ? 

69.  A  man  who  worked  for  $24  a  week  had  his  salary 
diminished  by  12^%  ;  what  was  it  after  the  deduction? 

70.  A  farmer  raising  300  bu.  of  wheat,  sold  66|%  of  it, 
and  fed  the  rest  to  his  stock ;  how  much  did  he  feed  ? 

71.  A  man  was  hired  to  work  80  days,  but  he  lost  20%  of 
his  time ;  how  many  days  did  he  work  ? 


104 


AMERICAN  MENTAL  ARITHMETIC. 


§  37.    Operation  to 

BE 

Determined. 

72. 

12  is  liow  many  times  6  ? 

83. 

18    is   how   many   hun- 

73. 

12  is  what  part  of  24? 

dredths  of  72  ? 

74. 

12  is  how  many  fifths  of 

84. 

18  is  what  %  of  72? 

24? 

85. 

4  is  what  %  of  2? 

75. 

12    is    how    many    hun- 

86. 

12  is  6  times  what  num- 

dredths of  24? 

ber? 

76. 

12  is  what  %  of  24? 

87. 

12  is  1  of  what? 

77. 

15  is  how  many  times  5  ? 

88. 

12  is  .06  of  what? 

78. 

15  is  what  part  of  45  ? 

89. 

12  is  6%  of  what? 

79. 

15  is  how  many  sevenths 

90. 

15  is  3  times  what? 

of  45? 

91. 

15  is  1  of  what  ? 

80. 

15    is    how    many    hun- 

92. 

15  is  .03  of  what? 

dredths  of  45  ? 

93. 

15  is  3%  of  what? 

81. 

15  is  what  %  of  45? 

94. 

24  is  |  of  what? 

82. 

18  is  how  many  sixths  of 

95. 

24  is  .06  of  what? 

72?  . 

96. 

24  is  6%  of  what? 

Ex.  72.   Since  12  is  some  number  times  6,  the  number  is  12  -~  6,  or  2. 

Ex.  73.   Since  12  is  some  number  times  24,  the  number  is  12  -4-  24,  or  \. 

Ex.  74.   Since  12  is  some  number  times  24,  the  number  is  12  -4-  24,  or  £ 
$  =  2}  fifths. 

Ex.  75.   Since  12  is  some  number  times  24,  the  number  is  12  -4-  24,  or  £ 
|  =  .50. 

Ex.  76.   Since  12  is  some  number  times  24,  the  number  is  12  -4-  24,  or  \ 
£  =  50%. 

Ex.  86.   Since  12  is  6  times  some  number,  the  number  is  12  -4-  6,  or  2. 

Ex.  87.   Since  12  is  §  times  some  number,  the  number  is  12  -=-  f ,  or  18. 

Ex.  88.   Since  12  is  .06  times  some  number,  the  number  is  12  -4-. 06,  or  200. 

Ex.  89.    Since  12  is  6  %  times  some  number,  the  number  is  12 -4- .06,  or  200. 

Note.  —  In  these  examples,  the  operation  must  be  determined  by  reason- 
ing. Each  admits  of  being  placed  in  the  form  of  an  equation,  and  the  oper- 
ation at  once  appears.  Thus,  Ex.  72,  12  =  some  no.  x  6.  Therefore,  the  no. 
=  12-4-6,  since  the  product  divided  by  either  factor  equals  the  other. 


PERCENTAGE.  10£ 

97.  A  boy  having  10^,  Ans.  50%.  Since  bf  is  some  num. 
lost  bt\  what  %  of  his  money  ber  of  times  10?>  the  number  is 
did  ho  lose?  *  +  **«»* 

98.  A   spent    60%    of    his  Ans.  $120.     He  spent  85  %  of  his 

£  -L  nzri   t       „  money  and  had  15%  left ;  since  15% 

money  for  a  horse,  2o%  for  a  /  .       *       >  /0 

at  o        /         t_  °^         money  is  $  30,  his  money  must 

saddle,  and  had  1 30  left;  what  be  #80 +  .16,  or  $200;  60%  of  $200 

did  he  pay  for  the  horse  ?  =  $  120. 

99.  Mary  had  12^  and  gave  3^  to  Henry;  what  %  of  her 
money  remained? 

100.  From  a  cask  containing  96  gallons  of  oil,  32  gallons 
were  drawn ;  what  %  of  the  whole  remained  in  the  cask  ? 

101.  A  teacher  whose  salary  is  $2400,  spends  $2000  annu- 
ally ;  what  °Jo  of  his  salary  does  he  save  ? 

102.  The  standard  of  gold  and  silver  coin  in  the  U.  S.  is 
9  parts  pure  gold  or  silver  and  1  part  alloy ;  what  %  is  alloy  ? 

103.  The  population  of  a  town  in  1892  was  1600 ;  in  1893 
it  was  2400 ;  what  was  the  °Jo  of  increase  ? 

104.  A  clerk  spends  $1200  a  year,  or  66|%  of  his  salary; 
what  is  his  salary  ? 

105.  A  man  drew  from  the  hank  $575,  or  25%  of  his 
deposit ;  what  was  his  deposit  ? 

106.  A  regiment  of  800  men  lost  160  men  in  battle ;  what 
(f0  -of  the  regiment  remained? 

107.  After  a  battle  80%  of  the  regiment,  or  640  men,  were 
left;  how  many  men  were  there  in  the  regiment  at  first? 

108.  A  spent  60%  of  his  money  for  a  horse,  25%  for  a 
carriage,  and  had  $60  left;  how  much  did  he  pay  for  the 
horse  ?     For  the  carriage  ? 

109.  The  master  of  a  ship  threw  overboard  800  bbl.  of 
flour,  or  16 J%  of  its  cargo;  what  was  its  cargo  at  first? 


106  AMERICAN  MENTAL   ARITHMETIC. 

110.  What  number  increased  by  5  times  itself  becomes  30  ? 

ill.  What  number  increased  by  §  of  itself  becomes  30? 

112.  What  number  increased  by  .06  of  itself  becomes  212? 

113.  What  number  increased  by  6%  of  itself  becomes  212? 

114.  What  number  increased  by  7  times  itself  becomes  40  ? 

115.  What  number  increased  by  |  of  itself  becomes  24  ? 

116.  What  number  Increased  by  16|%  of  itself  becomes  42? 

117.  What  number  increased  by  83J%  of  itself  becomes  22? 

118.  What  number  diminished  by  |  of  itself  becomes  30  ? 

119.  What  number  diminished  by  .06  of  itself  becomes  188? 

120.  What  number  diminished  by  6%  of  itself  becomes  188? 

121.  What  number  diminished  by  16| -%  of  itself  becomes  40? 

122.  What  nu mber  diminished  by  33  J%  of  itself  becomes  20  ? 

123.  What  number  diminished  by  66|  %  of  itself  becomes  20  ? 

124.  What  number  diminished  by  87^  %  of  itself  becomes  20  ? 

125.  What  number  diminished  by  8%  of  itself  becomes  184? 

Ex.  110.  A  number  increased  by  5  times  itself  becomes  6  times  itself ; 
since  6  times  the  number  is  30,  the  number  is  30  -f-  6,  or  5. 

Ex.  111.  A  number  increased  by  §  of  itself  becomes  f  times  itself ;  since 
\  times  the  number  is  30,  the  number  is  30  -r-  f ,  or  18. 

Ex.  113.  A  number  increased  by  6%  of  itself  becomes  106%  times  itself ; 
since  106%  times  the  number  is  212,  the  number  is  212  -f-  1.06,  or  200. 

Ex.  120.  A  number  diminished  by  6  %  of  itself  becomes  94  %  times  itself ; 
since-94%  times  the  number  is  188,  the  number  is  188  -^  .94,  or  200. 

In  these  examples,  what  the  %  is  of  is  not  given  ;  it  is  therefore  necessary 
to  assume  something  as  a  base.  As  in  the  former  case,  the  equation  enables 
us  to  determine  what  operation  to  perform.  Thus,  Ex.  114,  8  x  no.  =  40 ; 
the  no.  =  40  -^  8. 


PERCENTAGE.  107 

126.  A  merchant  sells  calico 

for    10%    more    this   year   than  Ans.  10?.    This  year's  price  is 

last  year  ;  this  year  he  sells  for  \10%  of  last  year's ;  since  110% 

.      /  J                       *  of   last  year's  price  is  11^,  last 

11?    a    yard;       what    was    last  year's  price  is  11^- 1.10,  or  Hy. 
year's  price  ? 

127.  A  merchant  sells  calico  Ans.  IOj*.    This  year's  price 

for  10%  less  this  year  than  last  is  90%  of  last  year's  Price  J  since 

year ;  this  year  he  sells  for  9^  a      90  %,of  last  year's  p™e  is  9/' last 
J       t        ,  ,  o      year's  price  is  9?  -f-  .90,  or  10^. 

yard ;  what  was  last  year  s  price  ? 

128.  By  running  15%  faster  than  usual,  a  locomotive  runs 
690  miles  a  day ;  what  is  the  usual  distance  it  runs  per  day  ? 

129. v  A  field  having  increased  in  productiveness  22%  over 
the  preceding  year,  yielded  488  bushels  of  potatoes;  what 
was  the  yield  the  previous  year? 

130. '  A  farmer  sold  1800  pounds  of  wool,  which  was  12*-% 
more  than  he  sold  the  previous  year ;  how  many  pounds  did 
he  sell  the  previous  year? 

131.  *  A  man  sold  20  cows  for  $400,  which  was  20%  less 
than  they  cost ;  what  did  they  cost  ? 

132.  After  paying  30%  of  his  debts,  a  merchant  found  that 
$  210  would  pay  the  remainder ;  what  did  he  owe  at  first  ? 

133.  My  salary  this  year  is  $75  per  month,  or  25%  more 
than  last  year ;  what  was  my  salary  last  year  ? 

134.  A  dealer  has  two  kinds  of  apples :  the  first  kind  he 
sells  for  66|%  more  than  the  second ;  he  sells  the  first  for 
$1.50  a  barrel;  for  how  much  does  he  sell  the  second  kind? 

135.  A  sells  a  cow  for  $49  which  is  12*-%  less  than  he 
gets  for  his  horse ;  what  does  he  get  for  the  horse  ? 

136.  I  pay  $12  per  week  for  board  this  year,  which  is 
20%  less  than  I  paid  last  year;  what  did  I  pay  last  year  ? 


108 


AMERICAN  MENTAL   ARITHMETIC. 


Names  are  sometimes  given 
to  the  terms  used  in  percentage. 

What  the  %  is  of,  is  the  base  ; 
the  base  x  the  %,  the  percent- 
age; base  +  percentage,  amount; 
base  —  percentage,  difference  ; 
the  %,  the  rate. 

To  find  percentage : 

137.  Base  60;  rate  7%. 

138.  Amount  80  ;  base  60. 

139.  Difference  70;  base  90. 

140.  Difference  40 ;  rate  20  % . 

141.  Amount  63;  rate  12-|-%. 

To  find  difference : 

146.  Base  32;  rate  25%. 

147.  Percentage  24 ;  base  30. 

148.  Base  72;  rate  33^%. 

149.  Percentage      25 ;      rate 

10%. 

150.  Percentage      30;      rate 

15%. 


Illustration. 

0%  of  200  =  12. 
200  +  12  =  212. 
200  -  12  =  188. 
200,  base. 

12,  percentage. 
212,  amount. 
188,  difference. 
C*%,rate%. 

To  find  amount : 

142.  Base  60;  rate  16f  %. 

143.  Percentage  30  ;  base  60. 

144.  Percentage  25  ;  rate  5  % . 

145.  Percentage    400 ;     base 
4000. 

To  find  rate  %  : 

151.  Base  60;  percentage  40. 

152.  Amount  70  ;  percentage 
30. 

153.  Difference   60 ;  percent- 
age, 20. 

154.  Amount  75  ;  base  60. 

155.  Difference  70  ;  base  80. 


To  find  base  : 

156.  Rate  6  %  ;  percentage  12. 

157.  Rate  6%  ;  amount  212. 

158.  Rate  6%;  difference  188. 

159.  Amount  90  ;  percentage 

10. 

160.  Rate  8%  ;  percentage  80. 


To  find  base : 

161.  Rate  8%  ;  amount  324. 

162.  Rate  8%  ;  difference  368. 

163.  Difference  70 ;   percent- 

age 30. 

164.  Rate  10%;    percentage 

70. 


PERCENTAGE.  109 

§  38.     Profit  and  Loss. 

What  is  the  selling  price? 

165.  Cost  $10;  gain  $2.  170.    Cost  $50;  loss  $5. 

166.  Cost  $36;  gain831%.      171.    Cost  $12;  loss  15%. 

167.  Cost  $24;  gain  121%.      172.    Cost  $20;  loss  66f%. 

168.  Cost  $25;  gain  40%.        173.    Loss  $16;  loss  8%. 

169.  Cost  $54;  gain  331%.      174.    Gain  $   9;  gain  75%?. 

What  is  the  cost  ? 

175.  Selling  price  4^;  gain  331%. 

176.  Selling  price  $21;  gain  $6. 

177.  Selling  price  $30;  gain  871%. 

178.  Selling  price  $20;  gain  25%. 

179.  Selling  price  $1.90;  loss  5%. 

180.  Selling  price  $25;  loss  $5. 

181.  Selling  price  4^;  loss  66|%. 

182.  Loss  5^;  loss  831%,. 

183.  Gain  $9;  gain  5%. 

What  is  the  gain  or  loss  %  ? 

184.  Selling  price  $12;  cost  $10. 

185.  Selling  price  $12;  gain  $3. 

186.  Selling  price  $12;  loss  $3. 

187.  Cost  $12;  gain  $3. 

188.  Cost  $12;  loss  $3. 

189.  Cost  $  25  ;  selling  price  $30. 

The  gain  or  loss  is  always  regarded  as  some  per  cent  of 
the  cost. 

Ex.  166.    The  gain  is  §  of  $36,  or  $30  ;  the  selling  price  is  §6G. 

Ex.  175.  100%  of  C  =  cost ;  33£%  of  C  =  gain  ;  133J%  of  C,  or  f  of  cost 
=  \f  ;  cost  =  \.<f>  -4-  f ,  or  3ft. 

Ex.  184.  The  gain  is  $2  ;  $2  =  no.  x  $10  j  no.  =  2  *  10,  or  20%. 


(S) 


AMERICAN  MENTAL  ARITHMETIC. 


190.  By  selling  a  horse  for  Ana.  $200.  100%  of  C=cost;  25% 
8250  I  gained  25%  ;  what  did  of  C=gain ;  125%  of  c  =  $250  >  cost 

*       .       6           .  o  =  #250  -*- 1-25  =  $  200. 

the  horse  cost  r 

191.  By  selling  a  horse  for  Ant  $300.  100  %  of  C= cost;  10% 
1270  I  lost  10%;  What  did  of  C  =  loss;  90%ofC  =  $270;  cost- 
, ,      ,                   ,  9  $  270  -T-  .90  =  $  300. 

the  horse  cost  f 


192.  What  must  be/fche  selling  price  of  tea  that  cost 
pound,  to  gain  20%  ^  fi^G 

193.  What  %  is  gained  on  an  article  bought  at  $  4.50  and 
sold  for  $6?    'bVgc/lt 

194.  A  grocer  sells  corn  at  a  profit  of  12^  a  bushel  and 
thereby  gains  20%  ;  what  is  the  cost?     pQ  ^ 

195.  Papers  were  sold  for  5fl  each  at  a  gain  of  25%  ; 
what  was  the  gain  on  4  papers  ?    fCy^ 

196.  By  selling  books  at  $  1.88  there  was  a  loss  of  6%  ;  what 
was  the  cost  ?  ^QJ)^ 

197.  Two  horses  were  sold  for  $99  each ;  on  the  first  there 
was  a  loss  of  10%  ;  on^ie  j^cond  a  .gain  of  10%  ;  what  was 
paid  for  both  horses  ?  ^ 

198.  Do  I  gain  or  lose  or/  thf?  sale  of  both  horses  in  ex- 
ample 197  ?     How  much  ?fff  *  ^J^Q^J. 

199.  A  merchant  by  selling  silks  for  $12  mote  than  they 
cost  gained  66%%  ;  what  was  the  selling  price  fjflJ 

200.  Find  the  v^ofit  on  land  that  cost  $200  and  was  sold 
at  a  gain  of  12%/%/  7 

201.  Find  the  selling  price  of  grain  that  cost  $  200  and 
was  sold  at  a  loss  of  9%. 

202.  Find  the  %  gained  on  oil  bought  at  12^  and  sold  at  14^. 

203.  A  watch  that  cost  $100  was  sold  at  a  gain  of  10%  ; 
what  was  the  gain  ?     What  was  the  selling  price  ? 


0  u 


PERCENTAGE.  HI 


204.  What  is  gained  per  cent  by  selling  coal  at  1 6  a  ton 
that  cost  1 5  a  ton  ? 

205.  A  horse  was  sold  at  a  loss  of  $50,  which  was  10%  of 
the  cost ;  what  was  the  cost  ? 

206.  A  watch  was  sold  for  $  240,  at  a  gain  of  20  %  ;  wh^t 
was  the  cost  ? 

207.  By  selling  a*  cow  for  $24  more  than  she  cost,  a  farmer 
gained  37|%  ;  what  was  the  selling  price? 

208.  What  must  be  the  selling  price  of  tea  that  cost  30^  a 
lb.,  in  order  to  gain  33^%  ? 

209.  A  boy,  by  selling  newspapers  at  h$  each,  gains  66%</0  ; 
what  do  they  cost  him  ? 

210.  A  boy,  by  selling  newspapers  at  5^  each,  gains  60%  on 
the  selling  price  ;  what  do  they  cost  him  ? 

211.  An  article  was  bought  for  $4  and  sold  for  $6  ;  what 
was  the  gain  per  cent  ?  What  per  cent  of  the  selling  price 
was  gained?  f  J 

212.  How  shall  I  mark  goods  that  cost  $  12,  so  as  to  gain 
16}*?    If 

213.  How  shall  I  mark  goods  that  cost  $/5,  so  that  I  can 
deduct  10%  from  the  marked  price,  and  yet  make  8%  on  the 
cost? 

214.  What  %  is  gained  on  goods  sold  at  double  the  cost  ?  ]Q  0 

215.  A  carpet  which  cost  $12  was  sold  for  $9;  what  was 
the  loss  %  ?  %  iT 

216.  How  must  goods  that  cost  $15  be  marked  so  as  to 
gain  121%  ?  \\f 

217.  How  shall  I  mark  goods  that  cost  $6,  so  that  I  can 
deduct  10%  from  the  marked  price,  and  make  5%  on  the  cost? 

218.  How  must  I  sell  goods  that  cost  $56  so  as  to  gain 
37  J%  ?   17 


112  AMERICAN  MENTAL  ARITHMETIC. 

§  39.   Commission. 
A  person  may  be  employed  to  buy  or  sell  for  another ;  the 
employer  is  the  principal;   the  other,  the  agent;   the  price 
paid  for  the  service,  the  commission  ;  the  amount  returned  to 
the  principal,  the  net  proceeds. 

Illustration.  —  A  farmer  takes  to  a  grocer  1(3  barrels  of  apples  to  be 
sold  at  $2  a  barrel,  agreeing  to  pay  him  10%  commission  for  selling  them. 
It  is  just  that  the  grocer  should  keep  10%  of  the  entire  sales.     Hence, 

If  an  agent  sells,  his  commission  is  some  per  cent  of 
the  sales. 

Illustration. — A  woolen  manufacturer  sends  his  agent  $1050  with 
'nstructions  to  buy  wool  after  deducting  his  commission  of  5%.  If  the  agent 
took  5%  of  $  1050,  he  would  take  5%  of  what  he  pays  for  the  wool,  and  also 
5%  of  what  he  keeps.  It  is  not  just  that  he  should  receive  5%  of  what  he 
keeps,  because  he  performs  no  labor  for  it,  but  he  is  entitled  to  5%  of  what 
he  pays  for  wool.     Hence, 

If  an  agent  buys,  his  commission  is  some  per  cent  of 
the  purchase. 

219.  Find  the  amount  of  sales  when  the  principal  receives 
$40,  at  a  commission  of  20%. 

Ans.  $  50.  The  commission  is  20%  of  the  sales  ;  100%  of  sales = the  sales  ; 
the  sales  —  the  commission  =  the  net  proceeds  ;  80%  of  sales  =  $40  ;  sales  = 
$ 40  h-  .80  =  $ 50.     Proof.    20%  of  $ 50  =  $  10  ;  $  50  -  $  10  =  $40. 

220.  An  agent  sells  10  bbl.  of  apples  at  $2  a  bbl. ;  what 
is  his  commission  at  10%. 

Ans.    $2.     His  commission  is  10%  of  the  selling  price,  or  10%  of  $20,  or  $2. 

221.  Find  the  rate  of  commission  when  $  2  is  paid  for  a 
sale  of  $8. 

Ans.  25%.  The  commission  is  some  per  cent  of  the  sales  ;  if  $  2  is  some 
number  times  $8,  the  number  is  2  h-  8,  or  25%.     Proof.   25%  of  $8  =  $2. 


PERCENTAGE.  113 


V?> 


222.  A  woolen  manufacturer  sends  his  agent  $  1050  to 
invest  in  wool  after  deducting  5%  commission;  what  is  the 
purchase  price  and  what  is  the  commission? 

Ans.  $1000;  $50.  5%  of  purchase  is  commission;  100%  of  purchase  is 
the  purchase;  105%  of  purchase  =  $  1050 ;  purchase  is  $1050 -=- 1.05,  or 
$ 1000  ;  the  commission  is  5%  of  $  1000,  or  $  50.  Proof.  5%  of  $  1000=$  50  ; 
$1000 +  $50  =  $1050. 

223.  Find  the  amount  of  sales  when  an  agent  receives  $4 

from  a  2%  commission. 

Ans.  $  200.  The  commission  is  some  per  cent  of  the  sales ;  2%  of  the  sales 
is  $4  ;  the  sales  are  $4  -4-  .02,  or  $200.     Proof.    2%  of  $200  =  $4. 

224.  Find  the  commission  on  the  sale  of  a  farm  for  $1000, 
at  3%. 

225.  Find  the  commission  on  the  purchase  of  a  mill  for 
$1000,  at  3%. 

226.  Find  the  commission  when  an  agent  receives  $  220  to 
be  invested  in  goods  after  deducting  his  commission  of  10%. 

227.  How  many  pounds  of  sugar,  at  8^  a  pound,  can  an 
agent  buy  for  $40.80  after  deducting  his  commission  of  2%  ? 

228.  Find  the  rate  of  commission  when  $2  is  paid  for  a 
sale  of  $10. 

229.  Find  the  amount  of  sales  when  a  commission  of  2% 
pays  the  agent  $8. 

230.  Find  the  commission  on  the  sale  of  a  house  for 
$20,000,  at  5%. 

231.  Find  the  rate  of  commission  when  an  agent  receives 
$5  for  a  sale  of  $200. 

232.  Find  the  commission  when  an  agent  receives  $  3360  to 
be  invested  in  goods  after  deducting  his  commission  of  12%. 

233.  Find  the  amount  of  sales  when  a  commission  of  6% 
pays  the  agent  $  600. 

AM.    MENT.    AS. — 8 


114  AMERICAN  MENTAL  ARITHMETIC. 

234.  Find  the  net  proceeds  from  the  sale  of  20  barrels  of 
sugar  at  $4,  commission  10%. 

235.  Find  the  amount  of  the  purchase,  when  an  agent 
invests  $440  in  sugar  after  deducting  his  commission  of  10%. 

236.  A  lawyer,  having  a  debt  of  $  7000  to  collect,  settles 
for  60%  ;  his  commission  is  1|%  ;  how  much  does  he  remit? 

237.  A  lawyer  collects  a  debt,  takes  2%  for  his  fee,  and 
remits  the  balance,  or  $490 ;  what  is  his  fee? 

238.  An  agent  receives  15050  with  which  to  purchase 
goods,  after  deducting  his  commission  of  1  %  ;  what  was  the 
cost  of  the  goods  ? 

239.  A  buys  corn  at  1J%  commission,  and  2J%  for  guar- 
anteeing payment;  if  the  whole  cost,  including  commission 
and  guaranty  is  1 416,  what  was  the  first  cost  of  the  corn? 

240.  My  agent  in  Paris  has  bought  for  me  16  bales  of 
calico,  each  bale  containing  50  pieces  of  30  meters  each,  at 
20^  a  meter;  his  commission  is  1%  ;  how  much  must  I  send 
him? 

241.  In  buying  shoes  at  a  commission  of  2J%,  an  agent's 
commission  was  $  25  ;  how  much  did  he  invest  ? 

242.  An  agent  sells  10000  lb.  of  sugar  at  8^  per  pound, 
and  receives  1-|%  commission;  he  pays  $10  for  freight; 
find  his  commission  and  the  net  proceeds. 

243.  An  agent  receives  $1010  with  which  to  buy  shoes 
and  pay  his  commission  of  1%  ;  what  does  he  pay  for  the 
shoes?     What  is  his  commission?    ' 

244.  D  bought  a  horse  for  $250,  paying  2%  of  the  cost 
for  commission,  and  2%  of  cost  for  traveling  expenses;  he 
sold  him  at  an  advance  of  10%  on  the  entire  cost,  including 
commission  and  expenses ;  how  much  did  he  gain  ? 


INTEREST. 


Money  paid  for  the  use  of 
money  is  interest ;  the  money 
loaned  is  the  principal;  the 
sum  of  the  principal  and  in- 
terest is  the  amount. 

There  are  three  conceptions 
of  interest : 

That  the  principal  alone 
bears  interest,  simple  interest. 

That  the  principal  and  the 
interest  on  the  principal  at 
the  end  of  each  year  bear  in- 
terest, annual  interest. 

That  the  principal  and  the 
interest  on  the  principal  at 
the  end  of  each  year,  and  all 
other  unpaid  interest  at  the 
end  of  each  year  bear  inter- 
est, compound  interest. 

Unless  otherwise  stated, 
simple  interest  is  always  un- 
derstood. The  rate  of  interest 
is  regulated  by  law;  when- 
ever interest  is  in  excess  of  the 
legal  rate,  it  is  called  usury. 


If  $  6  is  paid  for  the  use  of  $  100, 
$100  is  the  principal;  $6,  interest; 
$  106,  amount. 

Suppose  $  100  is  loaned  for  3  yr., 
at  6  %,  and  the  interest  remains  un- 
paid until  the  end  of  this  period. 

At  the  end  of  the  first  year,  $6 
interest  is  due  by  each  of  the  three 
conceptions. 

At  the  end  of  the  second  year,  by 
the  first  conception,  another  $6,  or 
$12  in  all,  is  due;  by  each  of  the 
second  and  third,  in  addition,  the 
interest  of  the  first  $6  for  1  yr. 
(36^),  or  $  12.36  in  all. 

At  the  end  of  the  third  year,  by  the 
first  conception,  another  $  6  is  due,  or 
$18  in  all;  by  the  second,  in  addi- 
tion, the  interest  of  the  first  $  6  for 
2  yr.  (72^),  and  the  interest  of  the 
second  $6  for  1  yr.  (36^),  or  $19.08 
in  all ;  by  the  third,  in  addition,  the 
interest  of  the  first  36^  for  1  yr. 
(2.16^),  or  $19.1016  in  all. 

For  1  yr.  the  simple,  annual,  and 
compound  interest  are  the  same  ;  for 
2  yr.,  the  annual  and  compound  in- 
terest are  the  same,  and  greater  than 
the  simple  ;  for  3  yr.  or  more,  each 
is  different,  the  order  of  magnitude 
being  compound,  annual,  simple. 

115 


116 


AMERICAN   MENTAL   ARITHMETIC. 


§  40.   Simple  Interest. 


1.  What  is  the  interest  of 
|1  for  lyr.  at  6%? 

2.  What  is  the  interest  of 
llfor  1  mo.  at  6%? 

3.  What  is  the  interest  of 
II  fori  da.  at  6%? 


The  interest  of  $  1  for  1  yr.  at  6% 
is  .06  of  $  1,  or  Of. 

Since  the  interest  of  $  1  for  12  mo. 
is  G^,  for  1  mo.  it  is  yV  of  6fi,  or  5  m. 

Since  the  interest  of  $  1  for  30  da. 
is  5  m.,  for  1  da.  it  is  -fa  of  5  m.,  or 
£  of  a  m. 


To  be  memorized. 

The  interest  of  $  1  for  1  yr.  at  6%  is  6? ;  for  1  mo., 
for  1  da.,  \  of  a  mill. 


What  is  the  interest  of : 

4.  llfor  2  yr.  at  6%? 

5.  $2  for  3  yr.  at  6%? 

6.  $4  for  5yr.  at  6%? 

7.  |1  for  2  mo.  at  6%? 

8.  |1  for  3  mo.  at  6%? 

9.  |1  for  5  mo.  at  6%? 


10.  |2  for  7  mo.  at  I 

11.  1 3  for  9  mo.  at  i 

12.  1 5  for  7  mo.  at  l 

13.  $1  for  4  da.  at  ( 

14.  |1  for  10  da.  at 

15.  |3  for  6  da.  at  ( 
■? 


re? 


1c? 


16.  14  for  8  da.  at  6% 

17.  |1  for  2  yr.  2  mo.  2  da.  at  6%  ? 

18.  1 10  for  3  yr.  18  da.  at  8%  ? 

19.  |30  for  4  yr.  8  mo.  at  6%  ? 

20.  1100  for  63  da.  at  10%? 

21.  |1  for  5  yr.  5  mo.  5  da.  at  10%  ? 

22.  |1  for  33  da.  at  12%? 

Ex.  18.  $2.44.  The  interest  of  $1  for  3  yr.  at  6%  is  $.18;  for  18  da., 
%  .003  ;  for  the  whole  time,  $  .183 ;  of  $  10,  10  times  $.183,  or  $  1.83  ;  at  8%, 
\  more,  or  $2.44. 

Ex.  20.  $  1.75.  The  interest  of  $  1  for  63  da.  at  6%  is  $.0105  ;  of  $  100, 
$  1.05 ;  at  10%,  f  more,  or  $  1.75. 


INTEREST.  117 

What  is  the  interest  of  $1: 

23.  For  3  yr.  at  4%  ?  32.  For  8  mo.  at  12%  ? 

24.  For  5yr.  at  7%?  33.  For  1  da.  at  7%  ? 

25.  For  8  yr.  at  9%?  34.  For  1  da.  at  8%  ? 

26.  For  2  yr.  at  5}%  ?  35.  For  3  da.  at  9%  ? 

27.  For  7  yr.  at  2%?  36.  For  7  da.  at  12%  ? 

28.  For  5  mo.  at  4%?  37.  For  5  da.  at  2%  ? 

29.  For  7  mo.  at  7%  ?  38.  For  2  yr.  5  mo.  at  7%  ? 

30.  For  9  mo.  at  4%?  39.  For  7  yr.  7  mo.  at  8%? 

31.  For  4  mo.  at  9%  ?  40.  For  8  yr.  5  mo.  at  6%  ? 

What  is  the  amount  of  $  1 : 

41.  For  2yr.  6  mo.  at  8%  ? 

42.  For  3  yr.  4  mo.  at  6%  ? 

43.  For  5  yr.  8  mo.  6  da.  at  6%  ? 

44.  For  1  yr.  10  mo.  24  da.  at  5%  ? 

45.  For  9  mo.  18  da.  at  9%  ? 

46.  For  3  mo.  12  da.  at  6%  ? 

47.  For  7  yr.  at  7%? 

48.  For  4  yr.  10  mo.  at  6%  ? 

49.  For  8  mo.  12  da.  at  3%? 

50.  For  7  mo.  6  da.  at  4%  ? 

51.  For  6  mo.  9  da.  at  6%? 

52.  For  4  mo.  24  da.  at  8%  ? 

53.  For  1  yr.  6  mo.  at  7%? 

54.  For  5  yr.  9  mo.  at  6%  ? 

55.  For  10  yr.  1  mo.  6  da.  at  2%  ? 

56.  For  12  yr.  10  mo.  3  da.  at  6%  ? 

Ex.  23.  12^.  When  years  alone,  months  alone,  or  days  alone  are  given, 
it  may  be  simpler  to  find  the  interest  directly  without  the  6  %  method.  Thus, 
the  interest  of  $  1  for  1  yr.  at  4  %  is  4  ^,  for  3  yr.  12  f. 

Ex.  41.    %  1.20.    The  amount  is  the  interest  plus  the  principal. 


69.  |40  in  4  mo.  at  uy0 

70.  |60  in  3  yr.  at  5% 

71.  |15  in  9  mo.  at  10% 


118  AMERICAN   MENTAL   ARITHMETIC. 

What  principal  will  gain  : 

57.  $24  in  4  yr.  at  6%  ?  61.  $63  in  3  yr.  at  3%  ? 

58.  $36  in  3  yr.  at  2%  ?  62.  $72  in  9  yr.  at  4%? 

59.  $48  in  6  yr.  at  4%  ?  63.  $20  in  2  yr.  at  5%  ? 

60.  $56  in  2  yr.  at  7%  ?  64.  $ 30  in  3  mo.  at  6%  ? 

65.  $32  in  4  mo.  24  da.  at  8%  ? 

66.  $330  in  6  mo.  18  da.  at  6%  ? 

67.  $100  in  5  mo.  30  da.  at  5%  ? 

68.  $120  in  10  mo.  at  12%? 

? 

?  . 
? 

72.  $30  in  10  mo.  at  6%? 

73.  $80  in  1  yr.  4  mo.  at  12%  ? 

74.  $50  in  2  yr.  6  mo.  at  10%  ? 

What  principal  will  amount  to  : 

75.  $496  in  4  yr.  at  6%  ?  84.  $396  in  4  yr.  at  8%  ? 

76.  $236  in  2  yr.  at  9%  ?  85.  $516  in  8  yr.  at  9%  ? 

77.  $  600  in  2  yr.  at  10%  ?  86.  $412  in  4  mo.  at  9%  ? 

78.  $  620-  in  3  yr.  at  8%  ?  87.  $  205  in  5  mo.  at  6%  ? 

79.  $ 226  in  2  yr.  2  mo.  at  6%  ?  88.  $254  in  9  yr.  at  3%  ? 

80.  $318  in  8  mo.  at  9%  ?  89.  $312  in  6  mo.  at  8%  ? 

81.  $ 540  in  3  yr.  6  mo.  at  10%?  90.  $416  in  4  mo.  at  12%  ? 

82.  $256  in  4  yr.  at  7%  ?  91.  $321  in  7  mo.  at  12%  ? 

83.  $345  in  3  yr.  at  5%  ?  92.  $436  in  18  mo.  at  6%  ? 

Ex.  57.  $  100.  Assume  $  1.  $  1  in  4  yr.  at  6  %  will  gain  24^  ;  it  will  take 
as  many  dollars  to  gain  $24  as  24^  is  contained  times  in  $24,  or  -$100. 

Ex.  75.  $400.  Assume  $1.  $  1  in  4  yr.  at  6%  will  amount  to  $1.24; 
it  will  take  as  many  dollars  to  amount  to  $496,  as  $1.24  is  contained  times 
in  $  496,  or  $  400. 


INTEREST.  119 


In  what  time  will : 


93.  flOO  gain  $24  at  6%  ?  99.    $125  gain  $20  at  2%  ? 

94.  $200  gain  $30  at  5%  ?        ioo.   $100  gain  $90  at  9%  ? 

95.  $150  gain  $36  at  4%  ?       101.   $250  gain  $50  at  10%  ? 

96.  $300  gain  $42  at  7%  ?       102.   $425  gain  $51  at  12%  ? 

97.  $400  gain  $64  at  8%  ?        103.    $325  gain  $39  at  3%  ? 

98.  $108  gain  $27  at  5%  ?       104.   $500  gain  $60  at  6%  ? 

105.  A  sum  gain  J  of  itself  at  6  %  ? 

106.  A  sum  gain  i  of  itself  at  8%  ? 

107.  A  sum  gain  |  of  itself  at  10%  ? 

108.  A  suln  gain  itself  at  8%  ? 

109.  A  sum  gain  ^  of  itself  at  12%  ? 

110.  A  sum  gain  3  times  itself  at  4%  ? 
ill.    In  what  time  will  $100  amount  to  $124  at  6%  ? 

112.  $50  to  $52  at  4%  ?  119.   $40  to  $88  at  12%  ? 

113.  $25  to  $40  at  6%  ?  120.   $50  to  $75  at  5%  ? 

114.  $40  to  $68  at  7%  ?  121.   $50  to  $53  at  6%  ? 

115.  $40  to  $72  at  8%  ?  122.   $30  to  $36  at  10%  ? 

116.  $80  to  $84  at  5%  ?  123.   $40  to  $76  at  9%  ? 

117.  $70  to  $84  at  2%?  124.   $50  to  $62  at  6%? 

118.  $30  to  $33  at  10%?  125.   $60  to  $84  at  4%  ? 

126.  In  what  time  will  a  sum  amount  to  twice  itself  at  6%  ? 

127.  In  what  time  will  a  sum  quadruple  at  10%  ? 

128.  In  what  time  will  a  sum  double  at  7%  ? 

Ex.  93.  4  yr.  Assume  1  yr.  $  100  in  1  yr.  at  6  %  will  gam  $6  ;  it  will 
take  as  many  years  to  gain  $  24  as  $  6  is  contained  times  in  $  24,  or  4  years. 

Ex.  105.  8  yr.  4  mo.  This  means,  '  in  what  time  will  $  12  (any  princi- 
pal) gain  $6  at  6%?  '     Assume  1  yr.,  etc. 

Ex.  111.  4  yr.  This  means,  •  in  what  time  will  $  100  gain  $  24  at  6  %  ?  ' 
Assume  1  yr.,  etc. 

Ex.  127.  30  yr.  This  means,  'in  what  time  will  $10  (any  principal) 
gain  $  30  at  10  %  ?  '     Assume  1  yr.,  etc. 


120  AMERICAN   MENTAL   ARITHMETIC. 

At  what  %  will : 

129.  $  100  gain  $24  in  4  yr.?  135.  $  175  gain  $70  in  2  yr.? 

130.  $ 200  gain  $98  in  7  yr.?  136.  $  800  gain  $96  in  8 yr.? 

131.  $  300 gain  $ 81  in  9  yr.?  137.  $ 600  gain  $  60  in  5  yr.? 

132.  $ 500  gain  $  75  in  5  yr.?  138.  $  190  gain  $76  in  4  yr.? 

133.  $ 400 gain $84 in  3 yr.?  139.  $ 250  gain  $35  in  7  yr.? 

134.  $  100  gain $36  in  6  yr.?  140.  $100  gain  $81in9yr.? 

141.  A  sum  gain  J  of  itself  in  8  yr.? 

142.  A  sum  gain  ^  of  itself  in  6  yr.? 

143.  A  sum  gain  |  of  itself  in  7  yr.? 

144.  A  sum  gain  itself  in  8  yr*.  ? 

145.  A  sum  gain  two  times  itself  in  10  yr.? 

146.  A  sum  gain  three  times  itself  in  5  yr.? 

147.  At  what  %  will  $100  amount  to  $124  in  4  yr.? 

148.  $40  to  $72  in  8  yr.?  154.  $40  to  $84  in  10  yr.? 

149.  $80  to  $88  in  5  yr.?  155.  $30  to  $39  in  6  yr.? 

150.  $20  to  $38  in  9  yr.?  156.  $20  to  $42  in  11  yr.? 

151.  $20  to  $48  in  7  yr.?  157.  $30  to  $48  in  12  yr.? 

152.  $80  to  $96  in  2  yr.?  158.  $20  to  $27  in  7  yr.? 

153.  $60  to  $96  in  3  yr.?  159.  $20  to  $29  in  9  yr.? 

160.  At  what  %  will  a  sum  amount  to  twice  itself  in  10  yr.  ? 

161.  At  what  %  will  a  sum  triple  in  20  yr.  ? 

162.  At  what  %  will  a  sum  double  in  10  yr.  ? 

163.  At  what  %  will  a  sum  quadruple  in  12  yr.? 

Ex.  129.  6  %.  Assume  1  %.  $  100  in  4  yr.  at  1%  will  gain  $  4 ;  it  will 
take  as  many  %  to  gain  $  24  as  $  4  is  contained  times  in  $>  24,  or  6  %. 

Ex.  141.  6J  %.  This  means,  ■  at  what  %  will  $  12  (any  principal)  gain  $  6 
in  8  yr.  ?  '     Assume  1%,  etc. 

Ex.  147.  This  means, '  at  what  %  will  $  100  gain  $  24  in  4  yr.  ? '  Assume 
1  %,  etc. 

Ex.  161.  10%.  This  means,  '  at  what  %  will  $12  (any  principal)  gain 
$  24  in  20  yr.  ? '     Assume  1  %,  etc. 


INTEREST.  121 

164.  Is  it  proper  to  reason 

thus:  "Since  $1  amounts  to  „„  Mkm    .„ 

r  *es-     Because  $5  will  amount  to 

11.06,  $5    will   amount   to  5      6  times  as  much  as  *  1. 
times     11.06,     or     $5.30"? 
Why? 

165.  Is  it  proper  to  reason  _  .   , 

i  o  •  dn  No*    Tne  am0llnt  ls  in  every  case 

thus:  "Since  $1  amounts  to      once  the  principal  plus  the  interest< 

$  1.06  in  1  yr.,  in  2  yr.  it  will  By  reasoning  as  at  the  left,  we  make 
amount  to  2  times  $1.06,  or  the  amount  twice  the  principal  plus 
$2.12"?      Why?  the  interest. 

166.  If  $60  amounts  to  $70  in  1  yr.,  what  will  it  amount 
to  in  2  yr.  ? 

167.  If  $40  amounts  to  $70  in  5  yr.,  what  will  it  amount 
to  in  10  yr.? 

168.  If  $60  amounts  to  $100  in  4  yr.,  what  will  it  amount 
to  in  8  yr.  ? 

169.  At  what  %  will  $200  gain  $56  in  4  yr.? 

170.  At  what  %  will  a  sum  triple  itself  in  40  yr.  ? 

171.  At  what  °Jo  will  a  sum  gain  3  times  itself  in  30  yr.  ? 

172.  What  principal  will  gain  $200  in  3  yr.  at  10%  ? 

173.  What  principal  will  amount  to  $  224  in  2  yr.  at  6  %  ? 

174.  In  what  time  will  $200  gain  $160  at  8%  ? 

175.  In  what  time  will  $200  amount  to  $256  at  7%  ? 

176.  In  what  time  will  a  sum  gain  3  times  itself  at  10%  ? 

177.  In  what  time  will  a  sum  quadruple  itself  at  2%  ? 

178.  What  is  the  interest  of  $100  for  2  yr.  6  mo.  at  8%  ? 

179.  How  many  dollars  in  6  yr.  at  3%  will  gain  the  interest 
of  $100  for  4  yr.  at  6%? 

180.  If  $30  amounts  to  $60  in  3  yr.,  what  is  the  rate  of 
interest  ?     The  amount  in  4  yr.  ?     The  interest  each  year  ? 


122 


AMERICAN  MENTAL   ARITHMETIC. 


When  the  time  does  not 
exceed  123  days,  a  modifica- 
tion of  the  6%  method  is  in 
general  use. 

Moving  the  decimal  point 
two  places  to  the  left  in  the 
principal,  gives  tlve  interest 
for  60  days  at  6  %. 


Since  the  interest  of  $  1  for  60  da. 
at  6%  is  If,  and  \f  is  .01  of  f  1, 
dividing  the  principal  by  100,  that 
is,  moving  the  decimal  point  two 
places  to  the  left,  will  give  the  in- 
terest for  60  days  at  6%. 


What  is  the  interest  of : 

181.  |125  for  60  da.  at  6%  ? 

182.  1250  for  60  da.  at  6%? 

183.  $313  for  60  da.  at  6%? 

184.  |400  for  63  da.  at  6%  ? 

185.  $100  for  93  da.  at  6%? 

186.  $200  for  33  da.  at  6%  ? 

187.  $100  for  63  da.  at  10%  ? 


194.  $100  for  33  da.  at  12%  ? 

195.  $100  for  93  da.  at  12%  ? 

196.  $200  for  93  da.  at    7%? 

197.  $100  for  63  da.  at 

198.  $200  for  33  da.  at 

199.  $500  for  63  da.  at 

200.  $800  for  33  da.  at 


7%? 
7%? 


188.  $100  for  93  da.  at  10%  ?    201.  $200  for  93  da.  at 


189.  $100  for  33  da.  at  10%  ? 

190.  $100  for  63  da.  at    8%? 

191.  $100  for  93  da.  at    8%  ? 

192.  $100  for  33  da.  at    8%? 

193.  $100  for  63  da.  at  12%  ? 


202.  $300  for  33  da.  at  10%  ? 

203.  $200  for  93  da.  at  10%  ? 

204.  $400  for  63  da.  at  10%  ? 

205.  $400  for  33  da.  at  10%  ? 

206.  $800  for  63  da.  at    8%  ? 


Ex.  184.  $4.20.  $4  (moving  the  decimal  point  two  places  to  the  left)  is 
the  interest  for  60  days  ;  ^  of  $4,  or  20^,  the  interest  for  3  days. 

Ex.  191.  $2.07.  $1  is  the  interest  for  60  days  at  6%;  \  of  $1,  or  50^, 
the  interest  for  30  days ;  ^  of  50/',  or  5^,  the  interest  for  3  days,  $1.55,  the 
interest  at  6  % ;  $1.55+1  of  $  1.55,  or  $2.07,  the  interest  at  8  %. 

Ex.202.  $2.75.  $3  is  the  interest  for  60  days  at  6%;  $1.50,  for  30 
days;  Yof  for  3  days;  $1.65,  the  interest  at  6%;  10%  is-1/  0f  6%;  $1.65 
x  10  x  J  =  $2.75,  the  interest  at  10%.  The  last  step  is  taken  by  moving  the 
decimal  point  one  place  to  the  right,  and  dividing  by  6. 


INTEREST. 


123 


§  41.     Trade  Discount. 


Merchants  and  manufac- 
turers usually  have  fixed  price 
lists  of  their  goods,  and  when 
the  market  varies  they  change 
the  rate  of  discount  instead  of 
changing  the  fixed  price.  The 
fixed  price  is  the  list  price; 
the  deduction  is  the  trade  dis- 
count, the  amount  paid,  the 
net  price. 

They  announce  their  terms 
upon  their  bill  heads  thus, 
"Terms  30  days  less  5%," 
etc.  In  addition  to  this,  they 
frequently  offer  an  additional 
discount  for  cash. 


Sold  a  bill  of  goods,  list 
price  |20,  on  4  mo.  at  5% 
discount,  and  deducted  10% 
for  cash;  what  was  the  net 
price  ? 

Ans.  $17.10.  This  means  that 
payment  was  not  due  for  4  mos.; 
that  5%  of  list  price  was  to  be  de- 
ducted because  of  the  condition  of 
the  market ;  and  that  an  additional 
discount  of  10%,  after  the  first  dis- 
count had  been  subtracted,  was  made 
for  cash. 

$20  less  5%  of  $20  =  $19;  $19 
less  10%  of  $19  =  $17.10. 


207.  Sold  a  bill  of  goods,  list  price  1 20,  on  3  mo.  at  10% 
discount,  and  deducted  5%  for  cash;  what  was  the  net  price? 

208.  Sold  a  bill  of  goods,  list  price  $  20,  on  3  mo.  at  5% 
discount,  and  deducted  10%  for  cash ;  what  was  the  net  price  ? 

209.  Compare  examples  20T  and  208.  Does  it  make  any 
difference  with  the  net  price  whether  the  discount  is  10% 
off  and  5%  for  cash,  or  5%  off  and  10%  for  cash? 

210.  Which  is  the  better  for  the  purchaser,  10%  off  and 
5%  for  cash,  or  5%  off  and  10%  for  cash? 

211.  Compare  examples  207  and  208  and  decide  whether 
any  account  should  be  taken  of  the  time  before  the  bill  is  due 
in  computing  the  net  price. 


124 


AMERICAN  MENTAL   ARITHMETIC. 


§  42.    True  Discount. 

The  true  present  worth  of 

a  long-time  note,  or  of  a  sum  .       „        .         .,    dbftA_     ,. 

*  '  Am.   Present  worth    $  200 ;   dis- 

of  money  due  a  long  time  in     count  $  12.    The  present  worth  is 

that  sum  which  put  at  interest  to- 
day will  amount  to  $212  in  1  yr. 
Assume  $1.  $1  in  1  yr.  at  6% 
amounts  to  $  1.06  ;  it  will  take  as 
many  dollars  to  amount  to  $212,  as 
$  1.06  is  contained  times  in  $212,  or 
$200.  The  discount  is  the  debt 
minus  the  present  worth,  or  $  12. 


advance,  is  that  sum  which, 
put  at  interest  now,  will 
amount  to  the  debt  at  the 
expiration  of  the  time. 

John  Smith  owes  me  $212 
a  year  from  to-day;  what  is 
the  present  worth,  money  at 
6%?     What  is  the  discount ? 


Find  the  present  worth  of : 

212.  |412  due  in  6  mo.,  int.  6%. 

213.  $324  due  in  8  mo.,  int.  12%. 

214.  $321  due  in  1  yr.,  int.  7%. 

215.  $430  due  in  1  yr.  6  mo.,  int.  5%. 

216.  $520  due  in  8  mo.,  int.  6%. 

217.  $340  due  in  1  yr.  4  mo.,  int.  10%. 

218.  $210  due  in  10  mo.,  int.  6%. 

219.  $590  due  in  2  yr.,  int.  9%. 

220.  $336  due  in  3  yr.,  int.  4%. 

221.  $644  due  in  5  yr.,  int.  8%. 

222.  $427  due  in  9  mo.,  int.  9%. 

223.  $324  due  in  1  yr.  4  mo.,  int.  6%. 

224.  $230  due  in  1  yr.  8  mo.,  int.  9%. 

225.  $300  due  in  4  yr.  2  mo.,  int.  12%. 

226.  $266  due  in  5  yr.  6  mo.,  int.  6%. 

Ex.  212.   $  400.  $  1  will  amount  to  $  1.03  in  6  mo.  at  6  % ;  it  will  take  as 
many  dollars  to  amount  to  $412  as  $  1.03  is  contained  times  in  $412,  or  $400. 


INTEREST.  125 


§  43.    Bank  Discount. 

The  true  method  of  finding  the  present  worth  of  a  short- 
time  note  is  to  find  that  sum  which  will  amount  to  the  face 
of  the  note  in  the  given  time  at  the  given  rate.  But  since 
the  interest  on  the  face  for  the  given  time  at  the  given  rate 
is  nearly  the  same,  more  easily  found,  and  to  the  advantage 
of  the  lender,  the  latter  method,  known  as  Bank  Discount,  is 
employed. 

The  payer  is  allowed,  on  all  notes,  three  days,  called  days 
of  grace,  for  payment,  after  the  note  becomes  due.     Hence, 

In  finding  bank  discount,  three  days  are  always  added 
to  the  time. 

In  this  set  the  days  of  grace  are  included. 

What  is  the  bank  discount  of : 

227.  1360  for  33  da.  at  6%  ?  239.  $240  for  63  da.  at  12%  ? 

228.  $240  for  63  da.  at  6%  ?  240.  $ 240  for  93  da.  at  12%  ? 

229.  $360  for  33  da.  at  10%  ?  241.  |160  for  63  da.  at  8%  ? 

230.  $360  for  63  da.  at  10%  ?  242.  $370  for  63  da.  at  4%  ? 

231.  $430  for  33  da.  at  6%  ?  243.  $280  for  33  da.  at  5%  ? 

232.  $520  for  63  da.  at  9%  ?  244.  $410  for  93  da.  at  6%  ? 

233.  $640  for  93  da.  at  6%  ?  245.  $190  for  63  da.  at  6%  ? 

234.  $150  for  33  da.  at  6%  ?  246.  $200  for  33  da.  at  6%  ? 

235.  $260  for  63  da.  at  6%  ?  247.  $900  for  33  da.  at  10%  ? 

236.  $830  for  93  da.  at  6%  ?  248.  $720  for  63  da.  at  8%  ? 

237.  $240  for  93  da.  at  6%  ?  249.  $650  for  93  da.  at  6%  ? 

238.  $240  for  33  da.  at  12%?  250.  $875  for  33  da.  at  4%  ? 

Ex.  227.  %  1.98.  The  interest  of  $  360  for  60  da.  at  6  %,  is  $  3.60  j  for  30 
da.,  $  1.80  ;  for  3  da.,  $  .18  ;  for  33  da.,  $  1.98. 


126 


AMERICAN  MENTAL   ARITHMETIC. 


§  44.    Stocks. 


To  raise  money  for  the  pros- 
ecution of  business  enterprises, 
stock  companies  are  often  formed. 
Shares  are  issued  with  a  face  value 
(par  value)  of  $100,  but  some- 
times this  is  made  $50,  or  $25. 

Shares  do  not  often  sell  for  their 
par  value,  but  for  more  (above  par), 
or  for  less  (below  par),  according 
to  the  success  of  the  enterprise. 

Earnings  (dividends)  are  paid 
at  certain  periods. 

Brokers  buy  and  sell  stock  for 
their  principals,  charging  some- 
thing (brokerage)  both  for  buy- 
ing and  selling. 


Illustration. 
On  a  mining  claim  in  Mexico, 
gold  and  silver  were  found  in 
such  abundance  that  a  stock 
company  was  organized.  They 
issued  100000  shares  with  a  face 
value  of  $  100  a  share,  but  sold 
them  all  for  $  2  a  share.  The 
earnings  at  the  end  of  the  first 
year  were  $660000,  and  a  divi- 
dend of  6%  was  declared.  After 
the  dividend,  the  stock  sold  for 
$110  a  share.  At  this  time 
James  Lyman  bought  of  John 
Fluker,  through  a  broker,  50 
shares,  paying  \%  brokerage, 
and  received  the  certificate  rep- 
resented. He  afterwards  sold 
the  shares  at  $75,  paying  \% 
brokerage. 


INTEREST. 


127 


On  this  certificate  $100  is 
the  par  value  of  each  share ; 
|2,  1110,  175  were  the  mar- 
ket values  at  different  times. 

The  par  value  is  $100 
-unless  otherwise  stated. 

The  brokerage  and  divi- 
dend are  some  %  of  the  par 
value. 

251.  What  was  the  par 
value  of  the  50  shares  when 
bought  by  John  Fluker? 
What  was  the  market  value  ? 

252.  What  was  the  par 
value  of  the  50  shares  when 
bought  by  James  Lyman? 
What  was  the  market  value  ? 

253.  How  much  stock  did 
John  Fluker  own? 

254.  How  much  stock  did 
James  Lyman  own? 

255.  What  was  the  income 
on  one  share  at  the  time  of 
first  dividend? 

256.  How  much  brokerage 
did  Lyman  pay  on  one  share 
when  he  bought  ?  When  he 
sold? 

257.  How  much  did  James 
Lyman  pay  per  share  ?i  How 
much  did  he  receive  ? 


This  understanding  saves  confu- 
sion, and  makes  it  unnecessary  to 
call  attention  to  the  par  value. 

The  brokerage  and  dividend  must 
be  some  per  cent  either  of  the  market 
value  or  of  the  par  value.  The  par 
value  never  changes,  the  market  value 
is  constantly  changing ;  hence  the 
former  is  selected. 


Ans.  Par  value,  $  5000 ;    market 
value,  $  100. 


Ans.  Par  value,  $5000;    market 
value,  $  5500. 


Ans.  $5000  stock. 


Ans.  $  5000  stock. 

Ans.  $6.  The  income  on  one 
share  was  6  %  of  $  100,  or  $  6. 

Ans.  $  \  when  he  bought,  $  \ 
when  he  sold.  The  brokerage  was 
\%  and  £%  of  $100. 

Ans.  He  paid  $110^.  $110+$|- 
brokerage.  He  received  $  74|.  $  75 
—  $  I  brokerage. 


128 


AMERICAN   MENTAL   ARITHMETIC. 


258.  Does  the  market  value 
affect  the  dividend  ? 

259.  Does  the  dividend  or 
the  market  value  in  any  way 
affect  the  brokerage  ? 

260.  What  is  the  dividend 
on  one  share  of  6%  stock 
bought  at  50  and  sold  at  60  ? 

261.  What  is  the  market 
value  of  16000  5%  stock? 

How  many  shares  in : 

262.  $500  5%  stock? 

263.  18000  3%  stock? 

264.  $1000  7%  stock? 

265.  11200  10%  stock? 

266.  $1100  6%  stock? 

Find  the  cost  of  : 

272.  $800  stock,  at  80,  brokerage  J? 

273.  $400  stock,  at  90,  brokerage  J? 

274.  $500  stock,  at  60,  brokerage  J? 

275.  $1200  stock,  at  50,  brokerage  -J? 

276.  $1000  stock,  at  70,  brokerage  -|  ? 

277.  $8100  stock,  at  80,  brokerage  J? 

278.  $7200  stock,  at  50,  brokerage  |? 

279.  $400  stock,  at  40,  brokerage  f  ? 

280.  $600  stock,  at  90,  brokerage  J? 

281.  $700  stock,  at  60,  brokerage  j? 

Ex.  262.   5  shares.    The  par  value  of  one  share  is  $  100  ;  $  500  is  the  par 
value  of  as  many  shares  as  $  100  is  contained  times  in  $  500,  or  5  shares. 
Ex.  272.   $  641.    The  entire  cost  of  one  share  is  $  801 ;  the  cost  of  8  shares 


Ans.  No.     The  dividend  is  some 
%  of  the  par  valio'. 

Ans.  No.    The  brokerage  is  some 
%  of  the  par  value. 


Ans.  $  6.    The  dividend  is  some  % 
of  the  par  value. 

Ans.  We  have  no  means  of  know- 
ing. 

267.  $15000  8%  stock? 

268.  $2000  9%  stock? 

269.  $1400  12%  stock? 

270.  $9900  3%  stock? 

271.  $2500  4%  stock? 


INTEREST.  129 

Find  the  net  proceeds  of : 

282.  $800  stock  sold  at  80J-,  brokerage  J. 

283.  $400  stock  sold  at  90,  brokerage  £. 

284.  $  500  stock  sold  at  50^,  brokerage  \. 

285.  $600  stock  sold  at  70,  brokerage  J. 

286.  $  1200  stock  sold  at  80,  brokerage  f . 

287.  $  1000  stock  sold  at  40,  brokerage  |. 

288.  $900  stock  sold  at  90|,  brokerage  |. 

289.  $1200  stock  sold  at  60 J,  brokerage  \. 

Ex.  282.  $  640.    The  net  proceeds  on  one  share  is  $  80 ;  on  8  shares,  8 
times  $80,  or  $640. 

Find  the  dividend  on  : 

290.  $800  4%  stock  bought  at  90. 

291.  $400  2%  stock  sold  at  80. 

292.  $500  6%  stock  sold  at  70 {. 

293.  $600  7%  stock  bought  at  50. 

294.  $1200  3%  stock  sold  at  80  J. 

295.  $1500  5%  stock  sold  at  60. 

296.  $2000  9%  stock  sold  at  89. 

Ex.  290.    $32.     The  dividend  on  one  share  is  $4  ;  on  8  shares,  8  times 

$4,  or  $32. 

How  many  shares  may  be  bought : 

297.  For    $800  at  79|,  brokerage  |? 

298.  For  $1000  at  49f,  brokerage  J? 

299.  For  $1200  at  59|,  brokerage  -J? 

300.  For  $1400  at  69^,  brokerage  f  ? 

301.  For    $700  at  19|,  brokerage  |? 

302.  For  $2000  at  39|,  brokerage  |? 

303.  For  $3000  at  59|,  brokerage  £? 

Ex.  297.   10  shares.    The  cost  of  one  share  is  $  80 ;  $  800  will  buy  as 
many  shares  as  $  80  is  contained  times  in  $  800,  or  10  shares. 

AM.    MENT.    AR. 9 


130  AMERICAN  MENTAL  ARITHMETIC. 

How  much  stock  gives  an  income : 

304.  Of  $200;  stock  4%  ?  309.  Of  18000;  stock  4%  ? 

305.  Of  $400;  stock  5%  ?  310.  Of  $5000;  stock  5%  ? 

306.  Of  $120;  stock  6%?  311.  Of  $1800;  stock  9%  ? 

307.  Of  $100;  stock  2%?  312.  Of  $2000;  stock  10%? 

308.  Of  $900;  stock  3%  ?  313.  Of  $4800;  stock  12%  ? 

Ex.  304.  $  5000  stock.  The  income  on  one  share  is  $  4  ;  it  will  take  as 
many  shares  to  yield  $  200  as  $  4  is  contained  times  in  $  200,  or  50  shares. 
50  shares  =  $  5000  stock. 

What  %  will  I  realize  on  my  investment : 

314.  When  6%  stock  is  bought  at  80? 

315.  When  5%  stock  is  bought  at  50? 

316.  When  8%  stock  is  bought  at  40? 

317.  When  9%  stock  is  bought  at  70? 

318.  When  4%  stock  is  bought  at  20? 

319.  When  3%  stock  is  bought  at  60? 

320.  When  7%  stock  is  bought  at  70? 

Ex.  314.  7 $  %.    One  share  costs  $  80  and  gains  $  6  ;  the  gain  %  is  $  6  -=-  $  80, 

or7i%. 

Find  the  price  of  a  4%  stock  : 

321.  To  equal  a    6%  stock  at  50. 

322.  To  equal  a    5%  stock  at  20. 

323.  To  equal  a    7%  stock  at  60. 

324.  To  equal  an  8%  stock  at  80. 

325.  To  equal  a    5%  stock  at  40. 

326.  To  equal  a  12%  stock  at  80. 

327.  To  equal  a  10%  stock  at  50. 

Ex.  321.  $33^.  One  share  costs  $50  and  gains  $6  ;  $1  gains  -fa  of  $6, 
or  12^ ;  if  $  1  gains  12^,  it  will  take  as  many  dollars  to  gain  $  4  as  12^  is 
contained  times  in  $4,  or  $33k 


PRACTICAL   EXERCISES. 


§  45.   At  the  Lumber  Yard. 

The  same  piece  of  lumber  may  be  called  by  different  names 
since  the  classification  is  not  exact.     The  following  may  be 
helpful :  — 
Lumber  —  wooden  building  material. 

I.   Boards —  1  in.  thick  —  less  if  specified. 

1.  Stock  boards  —  boards  of  uniform  width —  12  in. 

wide. 

2.  Fencing  —  6  in.  wide. 

3.  Flooring  —  matched  boards. 

4.  Siding  or  clapboards  —  Jin.  thick  —  thicker  at 

one  edge. 
II.    Dimension  Stuff. 

1.  Scantling  —  2  in.  to  4  in.  thick  —  3  in.  to  4  in. 

wide. 

2.  Joist  —  2  in.  thick  —  any  width. 

3.  Plank  —  2  in.  thick  —  wider  than  4  in. 

4.  Timber  —  thicker  than  2  in.  —  wider  than  4  in. 

III.  Foot  Stuff — sold  by  linear  foot. 

1.  Battens  —  for  covering  cracks. 

2.  Moulding  —  for  finishing. 

IV.  Laths  —  4  ft.  long,  1 J  in.  wide  —  50  to  a  bunch. 

V.  Shingles  —  4  in.  wide  —  250  to  the  bunch.  They  are 
not  of  uniform  width,  but  every  4  in.  is  reckoned  as 
one  shingle. 

131 ' 


132 


AMERICAN   MENTAL   ARITHMETIC. 


Lumber  is    sold    by   the   board 
foot. 

A  board  foot  is  the  equivalent       ^dzwft+iiJJt^ 
of  1  ft.  long,  1  ft.  wide,  and  1  in.      ^mmmM^kz^ 
thick.  1  board  ft. 


Inch  lumber  12  ft.  long  con- 
tains as  many  board  ft.  as  there 
are  inches  in  its  width. 


For  12  ft.  x  t^-  ft.  x  1  in. 
1  ft.  x  1  ft.  x  1  in. 


How  many  feet  of  lumber  are  there  in 


1.  18  in.,  12  ft.  board? 

2.  18  in.,  10  ft.  board? 

3.  18  in.,  6  ft.  board? 

4.  18  in.,  14  ft.  board? 

5.  18  in.,  18  ft.  board? 

6.  3  6  in.,  10  ft.  boards? 

7.  4  7  in.,  16  ft.  boards? 

8.  1  joist  2  x  4,  12? 

9.  1  joist  2  x  4,  16  ? 

10.  1  scantling  3  x  4,  12  ? 


n.  1  plank  2  x  10,  12? 

12.  1  plank  2  x  10,  16  ? 

13.  4  timbers  4  x  4,  20  ? 

14.  3  pieces  8  x  8,  24? 

15.  5  pieces  2  x  4,  12? 

16.  8  3  in.,  12  ft.  boards? 

17.  6  14  ft.  fencing? 

18.  6  pieces  4  x  6,  20  ? 

19.  10  6  in.,  12  ft.  siding? 

20.  10  scantling  3  x  4,  16  ? 


Ex.  2.  7.  An  8  in.  12  ft.  board  would  contain  8  ft,  ;  10  =  12  -  \  of  12  ; 
8  —  |of8  =  6|;  taking  the  nearest  whole  number,  7. 

Ex.  7.  37.  4  7  in.  boards  =  1  28  in.  board.  A  28  in.  12  ft.  board  would 
contain  28  ft.  ;  16 =12+* \  of  12  \  28  +  \  of  28  =  87$  ;  taking  the  nearest 
whole  number,  37. 

Ex.  9.  11.  This  is  read,  "1  joist  2  by  4,  16."  The  joist  is  2  in.  thick, 
4  in.  wide,  16  ft.  long. 

Ex.  13.  107.  4,  4  x  4  pieces  =1  64  in.  board.  A  64  in.  12  ft.  board 
would  contain  64  ft. ;  20  =  12  +  §  of  12  ;  64  + 1  of  64  =  106§  ;  taking  the 
nearest  whole  number,  107. 

*  Unless  otherwise  specified,  stuff  less  than  an  inch  thick  is  counted  as  an 
inch  thick. 


PRACTICAL   EXERCISES. 


133 


Dealers  sometimes  use  a  card  like  the  following,  carried  out 
for  a  great  variety  of  lengths  and  dimensions.  Usually,  as  in 
this  table,  fractions  of  a  foot  are  neglected. 

Lumber  Table. 


SIZE. 

12 

14  i  16 

18  |  20  |  22 

24 

26  28 

30  |  32  |  34 

36 

38 

40 

2x6 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36 

38 

40 

2x8 

16 

19 

21 

24 

27 

29 

32 

35 

37 

40 

43 

45 

48 

51 

53 

3x4 

4x6 

2x4 

2x3 

21.  Verify  the  results  in  the  1st  horizontal  line. 

22.  Verify  the  results  in  the  2d  horizontal  line. 

23.  Declare  the  results  for  the  3d  horizontal  line. 

24.  Declare  the  results  for  the  4th  horizontal  line. 

25.  Declare  the  results  for  the  5th  horizontal  line. 

26.  Declare  the  results  for  the  6th  horizontal  line. 


How  many  board  feet  are  there  in 


27.  4plk.,  2x4,  12? 

28.  2  timbers,  4x6,  40? 

29.  9  joists,  2  x  3,  16  ? 

30.  12bds.,  1x12,  16? 

31.  24  fencing  1  x  6,  16  ? 

32.  4  piece  stuff,  3x3,  12? 

33.  6  scantling,  2  x  4,  16  ? 


34.  3  timbers,  6x6,  40? 

35.  2  timbers,  8x8,  60? 

36.  3  joists,  2x8,  12? 
*37.  6plk.,  Ijxl2,  12? 

38.  6plk.,  Ifxl2,  12? 

39.  6plk.,  I|xl2,  12? 

40.  6plk.,  I|xl2,  12? 


*  Count  lumber  less  than  1  in.  thick  as  1  in. ;  from  1£  to  1\,  as  1| ;  from 
If  to  1£,  as  1£  ;  from  If  to  2,  as  2. 


134  AMERICAN  MENTAL  ARITHMETIC. 

Laths. 

Laths  are  nailed  upon  joists  !  of  an  inch  apart  to  receive 
tlie  plaster. 

41.  How  many  laths  are  there  in  a  bunch  ?     See  p.  131. 

42.  What  are  the  dimensions  of  a  lath  ? 

43.  How  many  sq.  in.,  including  the  space  between  two 
laths,  does  1  lath  cover  ? 

44.  How  many  sq.  in.  will  1  bunch  cover? 

45.  How  many  sq.  in.  are  there  in  3  sq.  yd.  ? 

46.  How  do  the  results  in  Exs.  44  and  45  agree  with  the 
contractor's  rule,  "One  bunch  of  laths  will  cover  3  sq.  yd."? 

47.  Using  contractor's  rule,  how  many  bunches  of  laths 
will  be  required  for  the  ceiling  of  a  room  18  ft.  x  18  ft.? 

Shingles. 
A  hunch  of  shingles  is  20   in.  wide  and  contains  25 
courses  on  each  side;  a  shingle  averages  J^  in.  wide. 

48.  How  many  shingles  are  there  in  a  bunch  ? 

49.  If  a  shingle  is  laid  4  in.  to  the  weather,  how  many  sq. 
in.  will  one  shingle  cover? 

50.  How  many  sq.  in.  will  1000  shingles  cover? 

51.  How  many  sq.  in.  are  there  in  100  sq.  ft.  ? 

52.  How  do  results  in  Exs.  50  and  51  agree  with  the 
carpenter's  rule,  "  One  thousand  shingles  laid  4  in.  to  the 
weather  will  cover  100  sq.ft.  "? 

53.  How  many  bunches  of  shingles  laid  4  in.  to  the  weather 
will  be  required  for  a  roof  20  ft.  x  40  ft.  ?  Use  contractor's 
rule. 

54.  How  many  bunches  of  shingles  laid  4  in.  to  the  weather 
will  be  required  for  a  roof  18  ft.  x  12  ft.  ? 


PRACTICAL   EXERCISES. 


135 


§  46.     Measurement  of  Logs. 


By  the  number  of 
board  ft.  in  a  log  is 
meant  the  number 
of  board  ft.  in  the 
largest  piece  of  tim- 
ber that  can  be  sawed 
from  the  log. 

A  piece  1  in.  x  1  in.  x  12  ft. 
(usually  written  1  x  1, 12)  contains 
1  board  ft. 


12.  feet 


4  Board  ft. 


The    area    of   the    greatest 
inscribed  square  is  AB  . 
AB1  +  AC2  =  BC2. 
(sq.  of  hyp.  =  sum  of  sqs.) 


2AB  =BC\ 

AB>  =  ^- 
2 


A  log  12  ft.  long  contains  as 
many  board  ft.  as  there  are  sq. 
in.  on  its  squared  end,  or  as  many 
board  ft.  as  there  are  sq.  in.  in 
half  the  square  of  its  diameter. 

If  the  log  is  to  be  sawed  into 
boards,  deduct  \  for  waste. 

How  many  feet  of  lumber  are  there  in  a  log : 

55.  Length  12  ft. ;  D.    8  in.  ?     58.  Length  24  ft. ;  D.  10  in.  ? 

56.  Length  16  ft. ;  D.  10  in.?     59.  Length  32  ft. ;  D.  18  in.  ? 

57.  Length  20  ft. ;  D.  12  in.?     60.  Length  16  ft. ;  D.  10  in.  ? 

Ex.  56.    67  ft.  of  lumber.     102  =  100 ;  \  of  100  =  50  ;  if  the  log  were  12  ft. 
long,  there  would  he  50  hoard  ft.;  50  +  \  of  50  =  67. 

How  many  feet  of  boards  are  there  in  a  log : 

61.  Length  12  ft. ;  D.    8  in.  ?     64.  Length  20  ft. ;  D.    6  in.  ? 

62.  Length  18  ft. ;  D.  10  in.?     65.  Length  36  ft. ;  D.  10  in.  ? 

63.  Length  24  ft. ;  D.  12  in.  ?     66.  Length  40  ft. ;  D.  16  in.  ? 
Ex.  61.    26  ft.    There  would  he  32  ft.  of  lumber  ;  32  -  \  of  32  =  26. 


13G  AMERICAN   MENTAL   ARITHMETIC. 

§  47.   At  the  Carpet  Store. 

Carpeting  is  usually  a  yard  wide,  and  sold  by  the  lineal 
yard.  It  is  cut  up  into  breadths,  and  these  breadths  are 
matched  and  sewed  together.  As  the  same  figure  is  repeated 
at  intervals  varying  from  1  inch  to  6  feet  according  to  the 
pattern,  few  carpets  can  be  matched  without  loss. 

Find  the  number  of  breadths  and  the  length  of  each 
breadth. 

A  room  14  ft.  by  13  ft.  is  to  be  carpeted. 

67.  How  many  breadths  will  be  required  if  they  run 
lengthwise?     How  many  if  they  run  crosswise?     Ans.  5. 

68.  If  there  is  no  loss  in  matching,  how  many  yards,  1  yd, 
wide,  should  be  purchased,  the  breadths  running  lengthwise  ? 
How  many  if  the  breadths  run  crosswise  ? 

69.  How  many  yards  are  turned  under  in  each  case  in 
Ex.  68? 

70.  If  the  same  figure  is  repeated  at  intervals  of  9  in., 
how  many  linear  inches  are  wasted  on  each  strip  running 
lengthwise  ?     Explain. 

71.  If  the  same  figure  is  repeated  at  intervals  of  9  in.,  how 
many  linear  inches  are  wasted  on  each  strip  running  cross- 
wise ?     Explain. 

72.  If  there  are  five  breadths,  on  how  many  is  there  waste 
in  matching  ?     Explain.     Ans.  4. 

73.  What  is  the  cost,  at  $2  a  yard,  breadths  running 
lengthwise,  the  same  figures  1  ft.  apart,  carpet  |  yd.  wide  ? 
What  is  the  cost,  breadths  running  crosswise? 

74.  14  ft.xl3  ft.  =  182  sq.  ft.;  182  sq.  ft.  =  20f  sq.  yd. 
Will  20|  yd.  of  carpet  1  yd.  wide  be  sufficient?     Why? 


PRACTICAL  EXERCISES.  137 

§  48.    With  the  Paper  Hanger. 

Wall  paper  is  sold  by  the  double  roll,  48  ft.  x  1|  ft.,  or  by 
the  single  roll,  24  ft.  x  1|-  ft.  It  is  cut  up  into  strips,  matched, 
and  pasted  upon  the  walls  or  ceiling. 

From  the  distance  around  the  room  in  feet,  deduct  3  ft. 
for  each  opening  {door  or  window).  The  remainder  -*-  1  will 
give  the  number  of  strips  required  for  the  walls. 

The  walls  and  ceiling  of  an  8  ft.  room  20  ft.  x  16  ft.  are  to 
be  papered ;  there  are  four  windows  and  a  door. 

75.  How  many  strips  for  the  walls  will  a  double  roll 
supply?     Explain. 

Ans.  6  strips.  There  is  a  loss  in  matching,  but  this  need  not  be  consid- 
ered, because  the  paper  does  not  extend  to  the  floor,  on  account  of  the  base 
board,  nor  to  the  ceiling,  on  account  of  the  border. 

76.  By  the  rule,  how  many  double  rolls  will  be  needed  for 
the  wall?     Explain.     Ans.  7. 

77.  If  the  strips  run  lengthwise,  how  many  strips  for  the 
ceiling  will  a  double  roll  supply?     Explain.     Ans.  2. 

78.  If  the  strips  run  crosswise,  how  many  strips  for  the 
ceiling  will  a  double  roll  supply  ?     Explain.     Ans.  3. 

79.  How  many  double  rolls  y  will  be  required  for  the  ceil- 
ing, if  the  strips  run  lengthwise  ? 

80.  How  many  double  rolls  will  be  required  for  the  ceil- 
ing, if  the  strips  run  crosswise  ? 

81.  Do  we  ever  in  practice  find  the  exact  number  of  sq. 
ft.  on  the  walls,  deduct  for  the  doors  and  the  windows,  and 
then  divide  by  the  number  of  sq.  ft.  in  a  double  roll  ?  Why 
not? 


138  AMERICAN  MENTAL  ARITHMETIC. 

§  49.    Average. 
What  is  the  average  of  80,  Ana.  78f .     Their   sum   is   630 ; 

85,  90,  70,  75,  100,  60,  70  ?         630  +  8  =  78*' 

A  better  way  is  to  begin,  with  the  first,  find  the  difference  between  it  and 
each  of  the  others,  and  unite  these  differences  as  we  proceed.  Thus, 
(85  -  80  =  5),  5  ;  (90  -  80  =  10  ;  add  to  5,  because  90  is  more  than  80),  15 ; 
(80  -  70  =  10  ;  subtract  10,  because  70  is  less  than  80),  5  ;  (80  -  75  =  5  ; 
subtract),  0;  (100-80  =  20;  add),  20;  (80-60  =  20;  subtract),  0; 
(80  —  70  =  10  ;  subtract),  —  10.  Since  the  total  of  the  differences  for  8  num- 
bers is  —  10,  the  average  difference  must  be  |  of  —10,  or— 1J  ;  80— 1J  =  78|. 

In  practice,  we  say  5,  15,  5,  0,  20,  0,  —  10,  —  1£,  78£. 

Find  the  average  of : 

82.  80,  70,  70,  60,  90,  84,  85,  87,  90,  93. 

83.  85,  86,  87,  88,  90,  92,  93,  94,  95,  96. 

84.  70,  75,  80,  85,  90,  95,  100,  90,  80,  70. 

85.  75,  80,  82,  84,  78,  85,  71,  60,  54,  96. 

86.  50,  80,  85,  70,  75,  100,  99,  86,  87. 

87.  76,  68,  54,  47,  97,  98,  96,  97,  95. 

88.  84,  83,  80,  87,  89,  88,  82,  81,  85. 

89.  66,  67,  52,  53,  47,  84,  87,  91,  90. 

90.  100,  100,  90,  95,  98,  99,  97, 100. 

91.  75,  76,  73,  50,  62,  64,  70,  80,  89. 

92.  91,  89,  94,  93,  91,  99,  100,  50,  60. 

93.  89,  89,  88,  87,  86,  93,  95,  96,  97. 

94.  45,  37,  29,  50,  49,  61,  62,  59,  48. 

95.  84,  64,  94,  74,  54,  100,  90,  80,  70. 

96.  65,  70,  80,  85,  90,  94,  86,  85,  72,  84. 

97.  90,  92,  92,  93,  93,  94,  94,  94,  96,  98. 

Ex.  83.  90.6.  It  is  best  to  take  as  the  base  some  number  ending  in  a 
cipher  ;  in  this  take  90.     -  5,  -  9,  -  12,  -  14,  -  12,  -  9,  -  5,  0,  6  ;  90.6. 


INVOLUTION  AND  EVOLUTION. 


A  number  written  to  the  right  of 
another,  a  little  above,  shows  how  many 
times  the  latter  is  used  as  a  multiplier. 

The  number  used  as  a  multiplier  is 
the  base;  the  number  showing  how  many 
times  the  base  is  used,  the  exponent;  the 
result,  the  power  ;  the  process,  involution. 

A  number  written  to  the  left  of  another 
in  the  symbol  •*/,  or  the  denominator  of 
a  fractional  exponent,  calls  for  the  base 
which  taken  this  number  of  times  as  a 
multiplier  will  produce  the  latter. 

The  result  is  the  root;  the  process, 
evolution. 

When  the  second  root,  usually  called 
the  square  root,  is  required,  the  figure  2 
is  not  written  in  the  -^/. 

The  third  root  is  usually  called  the 
cube  root. 


Illustration. 

24  =  16 ; 

read, 

2  to  the  4th  power  =  16. 

2,  base; 

4,  exponent; 

16,  power  ; 

means 

2x2x2x2  =  16. 

Vl6  =  2,  or  16*  =  2. 
4  calls  for  the  base  which 
taken  4  times  as  a  multi- 
plier will  produce  16 ; 
read, 
the  4th  root  o/16  =  2. 
Vl6  =  4; 
read, 
the  square  root  of  16  =  4. 

V8  =  2; 
read, 


the  cube  root  of  8  =  2. 

Declare  the  value  of :  Declare  the  value  of : 

1.  2*;  3*;  6*;  7*.  6.    Vl6;  V27;  V32;  VM. 

2.  83;  93;  53 ;  43.  Ifis-  27^-  32^-  81* 

3.  252;  232;  222;  21*.  ?>    *,'       »        j       . 

4.  172;  182;  192;  162.  8    27  >  1645  64*5  A 

5.  34;  26;  35;  23.  9.    125^;   64*;  36*;  27*. 

Ex.  8.,  Ans.  9.  This  means  extract  the  cube  root  of  27,  and  square  the 
result. 


PROPORTION. 


Division  may  be  expressed  by  writing 
the  dividend  before  and  the  divisor  after 
a  colon.  Such  an  expression  is  a  ratio. 
See  p.  44. 

The  dividend  is  the  antecedent;  the 
divisor,  the  consequent. 

Two  ratios  may  be  equal.  An  equality 
of  two  ratios  is  a  proportion. 

The  sign  of  equality  is  often  abbrevi- 
ated by  writing  only  the  extremities  of 
the  sign  '  =,'  making  : : 

The  first  and  last  terms  of  a  propor- 
tion are  extremes;  the  second  and  third, 
means. 

In  a  proportion,  the  product  of  the 
extremes  must  equal  the  product  of 
the  means. 

If  three  terms  of  a  proportion  are 
given,  the  other  may  be  found. 

The  mean  proportional  of  two  quanti- 
ties is  the  square  root  of  their  product. 

1.    What  is  the  value  of  the  ratio  27 :  81  ? 


Illustration. 

3  :  4,  ratio  ; 

read, 

3  is  to  4  ; 

means,  3-^-4. 

3,  antecedent. 

4,  consequent. 
3:4  =  0:8,  proportion; 

read, 
3  is  to  4  equals  6  is  to  8. 
3  :  4  :  :  6  :  8,  proportion ; 

read, 
3  is  to  4t  as  6  is  to  8. 

3  and  8,  extremes. 

4  and  6,  means. 
3  :  4  :  :  6  :  8. 

3x8  =  4x6. 

3  :  (  )  :  :  6  :  8. 

3x8 


()  = 


0 


4. 


12:()::():3. 
()  =  Vl2  x  3 


=  6. 
36 :  72  ?     48 :  144  ? 

2.  What  is  the  difference  between  24  -r-  3  and  27  :  9  ? 

3.  Which  is  the  greatest,  f ,  2  -f-  3,  or  1 :  2  ? 

4.  Find  the  missing  term  in  the  proportion  9  :  18  :  :  4  :  (  )  ;  in 
4  :  12  : :  (  )  :  2;  in  5  :  (  )  : :  7  :  14 ;  in  (  )  :  6  :  :  8  :  12. 

5.  What  is  the  mean  proportional  of  9  and  4  ?    20  and  5  ? 

140 


MENSURATION. 


§  50.  One  Dimension. 

That  which  has  one  dimension  is  a 
line. 

A  line  may  extend  in  the  same  direc- 
tion, a  straight  line  ;  or  it  may  constantly 
change  its  direction,  a  curved  line. 

If  two  straight  lines  in  a  plane  are  ex- 
tended, they  will  meet,  or  they  will  not 
meet.  If  they  do  not  meet,  they  are  par- 
allel; if  they  meet,  they  form  angles. 

If  two  lines  meet,  the  angles  will  be 
equal,  right  angles,  or  not  equal,  oblique 
angles  ;  the  larger  is  obtuse,  the  smaller, 
acute. 

A  straight  line  may  be  parallel  to  the 
horizon,  a  horizontal  line ;  perpendicular 
to  the  horizon,  a  vertical  line ;  or  neither 
parallel  nor  perpendicular  to  the  hori- 
zon, an  oblique  line. 


1.   Straight. 


2.  Curved. 


Parallel  Lines. 


Angles. 
1&2.  Right. 
3  &  4.    Oblique. 


3.  Obtuse. 

4.  Acute. 


/ 


1.  Horizontal  Line. 

2.  Vertical  Line. 

3.  Oblique  Line. 


Define : 
A  line. 

A  straight  line. 
A  curved  line. 
Parallel  lines. 


An  angle. 
A  right  angle. 
An  obtuse  angle. 
An  acute  angle. 


9.  Oblique  angles. 

10.  A  horizontal  line. 

11.  A  vertical  line. 

12.  An  oblique  line. 

141 


142 


AMERICAN  MENTAL  ARITHMETIC. 


§  51.    Two  Dimensions. 


That  which  has  two  dimensions  is  a 
surface. 

.    Straight  lines  may  inclose   a  plane 
surface,  a  polygon. 

The  least  number  of  straight  lines 
which  can  inclose  a  plane  is  three,  a 
triangle  (1).  The  three  lines  may  be 
equal,  an  equilateral  triangle  (2)  ;  two 
of  them  may  be  equal,  isosceles  tri- 
angle (3)  ;  or  no  two  of  them  equal,  a 
scalene  triangle  (4). 

A  triangle  may  have  one  right  angle, 
a  right-angled  triangle  (6) ;  one  obtuse 
angle,  an  obtuse-angled  triangle  (7)  ;  or 
three  acute  angles,  an  acute-angled  tri- 
angle (8). 

The  next  number  of  straight  lines 
which  can  inclose  a  plane  is  four,  a 
quadrilateral  (9).  The  quadrilateral 
may  have  both  pairs  of  its  opposite 
sides  parallel,  a  parallelogram  (10)  ;  one 
pair  parallel,  a  trapezoid  (11);  or  neither 
pair  parallel,  a  trapezium  (12). 

The  parallelogram  may  have  its  angles 
right  angles,  a  rectangle  (13) ;  or  not 
right  angles,  a  rhomboid  (14). 

The  rectangle  may  have  its  sides  all 
equal,  a  square  (15);  the  rhomboid  may 
have  its  sides  all  equal,  a  rhombus  (16). 


Illustration. 

The  face  of  this  page  is  a  plane 
surface,  or  a  plane. 

All  the  figures  on  this  page  are 
polygons. 


% 


^ 
K 


Triangles. 

2.  Equilateral. 

3.  Isosceles. 

4.  Scalene. 


kd 


Triangles. 

6.  Right-angled. 

7.  Obtuse-angled. 

8.  Acute-angled. 

/\i«r\rfi3-]--[i5i 

Quadrilaterals. 

10.  Parallelogram. 

11.  Trapezoid. 

12.  Trapezium. 

13.  Rectangle. 

14.  Rhomboid. 

15.  Square. 

16.  Rhombus. 


MENSURATION. 


143 


If  the  angles  of  a  polygon  are  equal, 
it  is  a  regular  polygon. 

A  regular  polygon  of  five  sides  is  a 
regular  pentagon  (17);  six  sides,  a  regu- 
lar hexagon  (18) ;  seven  sides,  a  regular 
heptagon,  etc. ;  infinite  number  of  sides, 
a  circle. 


Illustration. 


Regular  pentagon. 
Regular  hexagon. 
Circle. 


Define : 

13.  A  surface. 

14.  A  polygon. 

15.  A  triangle. 

16.  An  equilateral  triangle. 

17.  An  isosceles  triangle. 

18.  A  scalene  triangle. 

19.  A  right-angled  triangle. 


20.  An  obtuse-angled  triangle. 

21.  An  acute-angled  triangle. 

22.  A  regular  polygon. 

23.  A  regular  pentagon. 

24.  A  regular  hexagon. 

25.  A  regular  heptagon. 

26.  A  circle.  . 


27.  Beginning  with  plane  surface  (see  note),  define  parallelo- 
gram, rectangle,  rhomboid,  rhombus,  square. 

28.  Beginning  with  quadrilateral  (see  note),  define  parallelo- 
gram, rectangle,  square,  rhombus. 

29.  Beginning  with  parallelogram  (see  note),  define  square, 
rhombus. 

30.  Give  as  short  a  definition  as  possible  of  square,  rhombus. 

Note.  — A  definition  may  begin  with  different  terms,  e.g. : 

A  square  is  a  plane  surface  bounded  by  two  pairs  of  opposite  sides,  having 
each  pair  parallel,  having  its  angles  all  right  angles,  and  having  its  sides  all 
equal.     Or, 

A  square  is  a  quadrilateral  having  its  opposite  sides  parallel,  having  its 
angles  all  right  angles,  and  having  its  sides  all  equal.     Or, 

A  square  is  &  parallelogram  having  its  angles  all  right  angles  and  having 
its  sides  all  equal.     Or, 

A  square  is  a  rectangle  having  its  sides  all  equal. 

That  definition  is  the  best  which  is  the  shortest,  provided  it  begins  with  a 
term  which  is  understood  by  the  person  for  whom  the  definition  is  given. 


144 


AMERICAN  MENTAL   ARITHMETIC. 


B 
BC,  Base. 
AB  +  BC+AC,  Perimeter. 


AB  and  BC,  Legs. 
AC,  Hypotenuse. 


D 
CD,  Radius. 
AB,  Diameter. 


§  52.    Parts  of  a  Polygon. 

That  side  of  a  polygon  on  which 
it  is  supposed  to  rest  is  its  base; 
the  distance  around  a  polygon, 
its  perimeter;  the  perimeter  of 
a  circle,  its  circumference. 

In  a  right-angled  triangle,  the 
side  opposite  the  right  angle  is  the 
hypotenuse;  the  other  sides,  legs. 

In  a  circle,  a  line  from  the 
center  to  the  circumference  is  a 
radius;  a  line  passing  through 
the  center  and  bounded  at  both 
extremities  by  the  circumference 
is  a  diameter. 

The  altitude  of  a  triangle,  par- 
allelogram, or  trapezoid,  is  a  per- 
pendicular to  the  base,  from  the 
vertex  opposite  the  base. 

AB  is  the  altitude  in  each  of 
these  figures.  Observe  that  the 
base  must  sometimes  be  extended. 

31.  On  how  many  sides  may  a  triangle  be  supposed  to  rest  ? 

32.  How  many  bases  may  it  have  ? 

33.  How  many  altitudes  may  it  have  ? 

34.  Define  the  base  of  a  polygon. 

35.  Define  the  perimeter  of  a  polygon  ;  circumference  of  a  circle. 

36.  Define  the  radius  of  a  circle  ;  the  diameter. 

37.  Compare  the  following  with  the  definition  of  a  circle  de- 
veloped on  p.  143  :  A  circle  is  a  plane  surface  bounded  by  a  curved 
line,  every  point  of  which  is  equally  distant  from  a  point  within 
called  the  center. 


u 


c 

D    C 


D         B       E      D 
AB,  Altitude. 


MENSURATION. 


145 


§  53.     Rules. 


The  area  of  a  parallelogram 
is  the  product  of  its  base  and 
altitude. 

The  area  of  a  trapezoid  is  half 
the  product  of  its  altitude  and 
the  sum  of  its  parallel  sides. 

The  area  of  a  trapezium  is 
half  the  product  of  its  diagonal 
and  the  sum  of  the  perpendicu- 
lars from  the  vertices  to  the 
diagonal. 

The  area  of  a  triangle  is  half 
the  product  of  its  base  and  alti- 
tude. 

The  area  of  a  triangle  is  the 
square  root  of  the  continued 
product  of  the  half  sum  of  its 
sides  and  the  remainders  found 
by  subtracting  each  side  from 
the  half  sum  separately. 

The  square  of  the  hypotenuse 
of  a  right-angled  triangle  is 
equal  to  the  sum  of  the  squares 
of  the  other  two  sides. 

The  circumference  of  a  circle 
is  twice  the  radius  times  3.1416. 

The  area  of  a  circle  is  the 
square  of  the  radius  times 
3.1416. 

AM.  MENT.  AR.  —  10 


Illustration. 
Area  =  6  x  10  =  60  sq.  in. 


m 


O      I      If) 

Area  =  — i —  x  6  =  GO  sq.  in. 

2 


Area  =  ^-±_-  x  10  =  60  sq.  in. 

2  H 


Area  = 

6  x  10      OA 

=  30  sq.  in. 

2 

6  +  8  +10  _  ■  10 
2 
12-    6  =  6. 
12  -    8=4. 

XX 

12  -  10  =  2. 

Area  =  Vl2  x  6  x  4  x  2  =  24  sq.  in. 


52  =  32  +  42  or,  25  =  9  +  16. 


Circum.  =  2  x  10  x  3.1416 
=  62.832  in. 
Area  =  102  x  3.1416  =  314.16  sq.  in. 


146 


AMERICAN  MENTAL  ARITHMETIC. 


Applications. 

Find  the  circumference  of : 
48.  A  circle,  R  2  in. 


§  54 
Find  the  area  of : 

38.  A  rectangle,  B  6  in. ;  A  4  in. 

39.  A  rhomboid,  B12in.*;  A  9  in. 

40.  A  square,  B  13  in. 

41.  A  parallelogram,  B  14  in. ; 

A  7  in. 

42.  A  trapezoid,  II    sides  8,  10 

in. ;  A  6  in. 

43.  A  trapezium,  D  12  in. ;    J§ 

6,  8  in. 

44.  A  triangle,  sides  3,  4,  5  in. 

45.  A  triangle,  B  6  in. ;  A  8  in. 

46.  A  rhombus,  B  12  in. ;  A  9  in. 

47.  A  circle,  R  5  in. 

Find  the  altitude  of  : 

53.  A  tri.,  S  16  sq.  in. ;  B  2  in. 

54.  A  rect.,  S  40  sq.  in. ;  B  5  in. 

55.  A  trapezoid,    S  60  sq.  in. ; 

II  sides  4,  8  in. 

In  the  following  figures  find  the  part  indicated. 

59.  60.  61.  62 


Find  the  hypotenuse  of : 

49.  A  rt.  tri.,  sides  9, 12  in. 

50.  A  rt.  tri.,  sides  7,  24  in. 

Find  the  other  leg  of : 

51.  A  rt.  tri.,  hyp.  25  in.,  A  lb  in. 

52.  Art.  tri.,  hyp.  13 in., B  12 in. 

Note. — B,  base;  A,  altitude;  II, 
parallel ;  D,  diagonal ;  ±,  perpendicu- 
lar; R,  radius;  rt.  tri.,  right-angled 
triangle  ;  rect.,  rectangle  ;  S,  area. 

Find  the  base  of : 

56.  A  tri.,  S  24  -sq.  in. ;  A3  in. 

57.  A  parallelogram,  S  32  sq. 
in. ;  A  4  in. 

58.  A  square,  S  64  sq.  in. 


R  =  ? 


R  =  ? 


S  of  circle  =  ? 


Area  ring 


MENSURATION. 


147 


§  55.    Three  Dimensions. 

Illustrations. 


That  which  has  three  dimensions  is 
a  solid. 

That  part  on  which  a  solid  rests  is 
its  base;  its  other  surfaces  are  faces; 
the  union  of  two  faces,  an  edge;  the 
union  of  three  or  more  edges,  a  vertex. 

A  solid  may  have  two  bases  equal 
and  parallel  polygons,  and  its  faces 
rectangles,  a  prism.  If  its  bases  are 
triangles,  triangular  prism;  squares, 
square  prism;  .  .  .  circles,  circular  prism 
or  cylinder. 

A  solid  may  have  two  bases  parallel 
polygons,  and  its  faces  trapezoids,  frus- 
tum of  a  pyramid.  If  its  bases  are 
triangles,  frustum  of  triangular  pyra- 
mid; .  .  .  circles,  frustum  of  circular 
pyramid  or  frustum  of  cone. 

A  solid  may  have  one  base  and  its 
faces  triangles,  a  pyramid.  If  its 
base  is  a  triangle,  triangular  pyramid ; 
square,  square  pyramid;  .  .  .  circle, 
circular  pyramid  or  cone. 

A  solid  may  have  all  of  its  surfaces 
equal  and  regular  polygons  ;  four  tri- 
angles, tetrahedron;  eight  triangles, 
octahedron;  twenty  triangles,  icosahe- 
dron;  six  squares,  cube  ;  twelve  penta- 
gons, dodecahedron ;  an  infinite  number 
of  infinitely  small  polygons,  sphere. 


A- BCD,  Solid. 
BCD,  Base. 
ACD,  Face. 
AC,  Edge. 
A,  Vertex. 


A,  Pentagonal  prism. 

B,  Cylinder. 


A,  Frustum  of  hexagonal  pyramid 

B,  Frustum  of  cone. 


A,  Pentagonal  pyramid. 

B,  Cone. 


A,  Tetrahedron. 
C,  Icosahedron. 


148 


AMERICAN  MENTAL   ARITHMETIC. 


§  56.   Rules. 


The  convex  surface  of  a 
prism  or  cylinder  is  the  prod- 
uct of  the  perimeter  of  its 
base,  by  its  altitude. 

The  volume  of  a  prism  or 
cylinder  is  the  product  of  the 
area  of  its  base,  by  its  altitude. 

The  convex  surface  of  a 
pyramid  or  cone  is  half  the 
product  of  the  perimeter  of 
its  base,  by  its  slant  height. 

The  volume  of  a  pyramid 
or  cone  is  one  third  the  prod- 
uct of  the  area  of  its  base,  by 
its  altitude. 

The  convex  surface  of  the 
frustum  of  a  pyramid  or  cone 
is  half  the  product  of  the 
sum  of  the  perimeters  of  its 
two  bases,  by  its  slant  height. 

The  volume  of  the  frustum 
of  a  pyramid  or  cone  is  one 
third  the  product  of  the  sum 
of  the  areas  of  its  upper  base, 
lower  base,  and  mean  propor- 
tional base,  by  its  altitude. 

The  surface  of  a  sphere  is 
four  times  the  square  of  its 
radius  times  3.1416. 

The  volume  of  a  sphere  is 
four  thirds  times  the  cube  of 
its  radius  times  3.1416. 


G 
V       4 


s=2(3+4)  x6. 
v=3x4x6. 


Illustration. 

ICE* 


s=2x2 

x  3. 1416x6. 
t?=22x  3. 1416x6. 


s=i(16  +  16  +  16      s=  1(2x8x3.1416) 
+  16)xl0.  xlO. 

v=U16xlQ)x6.     tf  =  K8x8 

x  3.1416)  x  6. 


s=i(8  +  8+8+8        s=K2x4x3.1416 

+  16  +  16+16    +2x8x3.1416)x5. 
+  16)x5. 


(82+162 


v=  i(42x  3.1416 


+  V82xl62)  x 3.  +82x3.1416 

+  \/42x82x3.14162) 
x3. 

s=4x52x  3.1416. 
v=fx53x  3.1416. 


MENSURATION.                                      149 

Define : 

67.  Solid. 

75.  Regular  octahedron. 

68.  Prism. 

76.  Regular  dodecahedron. 

69.  Cylinder. 

77.  Regular  icosahedron. 

70.  Pyramid. 

78.  Sphere. 

71.  Cone. 

79.  Triangular  prism. 

72.  Frustum  of 

pyramid. 

80.  Pentagonal  pyramid. 

73.  Frustum  of 

cone. 

81.  Frustum  of  hexagonal  pyra- 

74. Regular  tetrahedron. 

mid. 

State  the  rule  f oi 

f convex  s 

surface  of : 

82.  A  prism. 

83.  A  cylinder. 

84.  A  pyramid. 

85.  A  cone. 

State  the  rule  for  volume  of: 

89.  A  prism. 

90.  A  cylinder. 

91.  A  pyramid. 

92.  A  cone. 

Find  the  convex  surface  of : 

96.  A  triangular  prism,  each 
side  of  base  3  in.,  altitude  12  in. 

97.  A  pentagonal  pyramid, 
each  side  of  base  2  in.,  slant 
height  8  in. 

Find  the  volume  of : 

100.  A  cylinder,  radius  of 
base  5  in.,  altitude  10  in. 

101.  A  hexagonal  pyramid, 
area  of  base  36  sq.  in.,  alti- 
tude 12  in. 


86.  A  frustum  of  a  pyramid. 

87.  A  frustum  of  a  cone. 

88.  A  sphere. 


93.  A  frustum  of  a  pyramid. 

94.  A  frustum  of  a  cone. 

95.  A  sphere. 


98.  Frustum  of  a  square 
pyramid,  one  side  of  upper  base 
5  in.,  one  side  of  lower  base  10 
in.,  slant  height  12  in. 

99.  A  sphere,  radius  6  in. 

102.  Frustum  of  a  cone,  radi- 
us of  upper  base  5  in.,  of  lower 
base  10  in.,  altitude  12  in. 

103.  A  sphere,  radius  3  in. 


150 


AMERICAN   MENTAL   ARITHMETIC. 


§  57.     Similarity. 


Similar  figures  must  fulfill  two 
conditions : 

1st.  For  every  angle  of  the 
one  there  must  be  an  equal 
angle  in  the  other. 

2d.  The  sides  about  the 
equal  angles  must  be  in  pro- 
portion. 

In  similar  figures : 

Linear  parts  are  to  each 
other  as  homologous  linear 
parts. 

Surfaces  are  to  each  other 
as  the  squares  of  homologous 
linear  parts. 

Volumes  are  to  each  other 
as  the  cubes  of  homologous 
linear  parts. 

104.  The  radii  of  two  spheres 
are  4  in.  and  2  in.  What  is  the 
ratio  of  their  circumferences  ? 

105.  What  is  the  ratio  of  their 
surfaces  ? 

106.  What  is  the  ratio  of  their 
volumes  ? 


Illustration. 


Oo 

Similar  figures. 
CF:cf::DC:dc. 
12    :  6  :  :     4    :  2. 


Area  DF :  Area  df;  :  CF2  :  c/2. 
Area  DF:  Area  df:  :   122  :   62. 

Vol.  AF:  Vol.  af:  :  CF*  :  c/3. 
Vol.  AF:  Vol.  af:  :  123    :    63. 

Cx  :  C2  :  :  4  :  2. 
2  s  1,  Arts. 

SX:S2::&:  22. 
4  :  1,  Ans. 


Vx :  V2 :  :  43  :  23. 
8:1,  Ans. 

107.  The  circumference  of  a  lead  pipe  is  6  in. ;  what  is  the 
circumference  of  a  pipe  whose  diameter  is  half  the  diameter  of 
the  first  ? 


MENSURATION.  151 

108.  The  area  of  a  circle  is  10  sq.  in.;  what  is  the  area  of  a 
circle  whose  diameter  is  twice  the  diameter  of  the  first  ? 

109.  Two  lead  pipes  are  1  in.  and  2  in.  in  diameter.  The  area 
of  a  horizontal  section  of  the  one  is  how  many  times  a  similar 
section  of  the  other  ? 

110.  How  many  lead  pipes  1  in.  in  diameter  will  discharge  as 
much  water  as  one  pipe  4  in.  in  diameter  ? 

111.  A  cannon  ball  weighs  32  lb.;  what  is  the  weight  of  a 
similar  ball  whose  diameter  is  half  the  diameter  of  the  first  ? 

112.  What  is  the  ratio  of  the  surfaces  of  the  two  balls  in  Ex.  Ill  ? 

113.  A  is  6  ft.  tall ;  his  bronze  statue  is  12  ft.  tall ;  if  the  length 
of  A's  little  finger  is  2\  in.,  what  is  the  length  of  the  little  finger 
of  the  statue  ? 

114.  If  it  costs  $  1  to  paint  a  statue  of  A's  size,  what  will  it  cost 
to  paint  the  statue  in  Ex.  113  ? 

115.  If  a  statue  of  A's  size  weighs  500  lb.,  what  will  the  statue 
in  Ex.  113  weigh  ? 

116.  If  a  bin  6  ft.  deep  holds  60  bu.,  what  is  the  contents  of 
a  similar  bin  12  ft.  deep  ? 

117.  If  it  costs  $10  to  make  an  excavation  6  ft.  deep,  what  is 
the  approximate  cost  of  a  similar  excavation  24  ft.  deep  ? 

118.  If  it  costs  $1200  to  build  a  house  20  ft.  by  30  ft.,  what 
will  be  the  approximate  cost  of  a  similar  house  30  ft.  by  45  ft.  ? 

119.  If  it  costs  $  16  for  material  and  labor  to  lay  a  floor  16  ft. 
by  20  ft.,  what  will  it  cost  approximately  to  lay  a  similar  floor 
20  ft.  by  25  ft.  ? 

120.  If  it  costs  $  40  to  paint  a  house  30  ft.  by  40  ft.,  what  will 
it  cost  approximately  to  paint  a  similar  house  45  ft.  by  60  ft.  ? 

121.  Four  pipes  each  2  in.  in  diameter  empty  into  a  tank ;  what 
must  be  the  diameter  of  a  single  pipe  to  carry  away  all  of  the  water? 

122.  A  and  B  bought  a  ball  of  twine  8  in.  in  diameter  for  $  1 ; 
A  wound  from  the  outside  until  the  diameter  of  the  ball  that  was 
left  was  4  in. ;  what  should  each  pay  ? 


MISCELLANEOUS. 


§  58.    Arithmetical  Progression. 


A  series  of  numbers  may  increase  or 
decrease  by  a  common  difference,  an 
arithmetical  progression. 

The  first  term  is  written,  a;  the 
last  term,  I',  the  number  of  terms,  n\ 
the  common  difference,  d ;  and  the  sum 
of  the  terms,  s. 

Every  problem  may  be  solved  by  the 
formulae : 

Z=a  +  (tt-l)d  (1) 


-=(«+<> 


(2) 


Illustration. 
3,      5,     7,    9,  11 
14,     11,    8,    5,   .3 
arithmetical  progressions. 
Formula  (1).     The  last 
term  equals  the  first  term, 
plus  the  number  of  terms 
less  one  times  the  common 
difference. 

Formula  (2).  The  sum 
of  the  terms  equals  half  the 
number  of  terms  times  the 
sum  of  the  first  and  last 
terms. 


Ans.  19;  1=   7+4x3  =  19. 
Ans.  65;  g=§  (7  +  19)  =05. 


The  arithmetical  mean    of  two 
numbers  is  half  their  sum. 

1.  Find  I  when  a  =  7,  n  =  5,  d  =  3. 

2.  Find  s  when  n  =  5,  a  =  7,  /  =  19. 

3.  State  the  series  in  Ex.  1.     Prove  the  answer  to  Ex.  2. 

4.  Find  the  arithmetical  mean  between  7  and  19. 

5.  Find  the  sum  of  the  numbers  1  to  25  inclusive. 

6.  Find  the  sum  of  the  numbers  1  to  99  inclusive. 

7.  Translate  each  formula  for  arithmetical  progression. 

8.  How  far  can  a  man  walk  in  10  days,  going  10  miles  the 
first  day  and  increasing  the  rate  5  miles  per  day  ? 

152 


MISCELLANEOUS. 


153 


§  59.    Geometrical  Progression. 


A  series  of  numbers  may  increase  or 
decrease  by  a  common  ratio,  a  geometri- 
cal progression. 

The  first  term  is  written,  a;  the 
last  term,  I;  the  number  of  terms,  n; 
the  ratio,  r ;  and  the  sum  of  the  terms,  s. 

Every  problem  may  be  solved  by  the 

formulae : 

I  =arn-1;   (1) 
s  =  rl-a    (2) 
r  —  1 
The    geometrical    mean    of  two 
numbers  is  the  square  root  of  their 
product. 

9.  Find  I  when  a  =  2,  r  =  5,  n  =3. 

10.  Find  s  when  r  =  5,  I  =  50,  a  =  2. 


Illustration. 

2,       6,     18,     54 

64,     32,     16,       8 

geometrical  progressions. 

Formula  (1).    The  last 

term  equals  the  first  term, 

times  the  ratio  raised  to 

the  power  denoted  by  the 

number  of  terms  less  one. 

Formula  (2).     The  sum 

of  the   terms    equals    the 

quotient,   whose    dividend 

is  the  ratio  times  the  last 

term,  less  the  first  term; 

and  whose  divisor  is  the 

ratio  less  one. 

1=     2x52=50. 
5x50-2 


Ans.  50 
Arts.  62 


-=62. 


•5-1 

11.  State  the  series  in  Ex.  9.     Prove  the  answer  to  Ex.  10. 

12.  The  extremes  are  2  and  250 ;  the  ratio  is  5 ;  find  s. 

13.  A  man  bought  6  yards  of  cloth,  giving  2  4  for  the  first  yard, 
6  $  for  the  second,  18  4  for  the  third,  and  so  on ;  what  did  he  pay 
for  the  last  yard  ?     What  did  he  pay  for  all  ? 

14.  What  is  the  geometrical  mean  between  4  and  25  ? 

15.  State  the  two  formulse  for  geometrical  progression ;  trans- 
late each. 

16.  A  man  sold  a  pair  of  horses,  receiving  %  1  for  the  first  shoe, 
%  2  for  the  second,  %  4  for  the  third,  and  so  on ;  the  horses  being 
fully  shod,  how  much  did  he  receive  ? 

17.  A  man  bought  a  pair  of  oxen,  paying  1  f  for  the  first  shoe, 
2^  for  the  second,  4^  for  the  third,  and  so  on;  how  much  did  he 
pay  for  the  last  shoe,  the  oxen  being  fully  shod  ? 


154  AMERICAN  MENTAL  ARITHMETIC. 

§  60.  Specific  Gravity. 

The  weight  of  a  substance  divided  Illustration. 

by  the  weight  of  an  equal  volume  of  a  vol.  lead  weighs   ...  22  lb. 

water  is  its  specific  gravity.  Same  vol.  water  weighs     2  lb. 

Approx.  Table,  S.  G.  s#  g#  lead  _  22  1b.  _  n 
Gold,     19          Glass,      3          Water,     1  2  lb- 
Lead,    11          Stone,     3          Acid,     1.8  Gold  is  19  times  as  heavy- 
Silver,  10          Oak,       .7          Oil,         .9  as  water. 
Iron,       7          Cork,     .2          Air,         .001  Oak  wood  is  .7  as  heavy  as 

A  pint  is  a  pound  the  world 
round. 

A  cubic  foot  of  water  weighs  a  pint  of  water  weighs  a 

62.5  lb.  pound  (approx.). 


water. 

Air    is    .001    as    heavy  as 
water. 


18.  What  is  the  weight  of  a  pint  of  gold?  Of  lead?  Of  air? 

19.  What  is  the  weight  of  a  cubic  foot  of  cork  ? 

20.  What  is  the  weight  of  a  gallon  of  water  ?  Of  oil  ?  Of  acid  ? 

21.  What  is  the  weight  of  a  bushel  of  cork  ? 

22.  What  is  the  weight  of  a  cu.  ft.  of  oak  wood  ? 

23.  Two  volumes  of  lead  and  3  of  the  same  volumes  of  water 
weigh  25  oz. ;  what  is  the  specific  gravity  of  lead  ?     Explain. 

24.  Of  a  mixture,  \  in  volume  is  oil  and  -§  water ;  what  is  the 
S.  G.  of  the  mixture  ? 

25.  What  is  the  S.  Gr.  of  lead  and  silver  compounded  of  equal 
volumes  ? 

26.  How  many  cu.  ft.  of  cork  will  weigh  as  much  as  a  cu.  ft. 
of  lead? 

27.  What  is  the  difference  in  lb.  between  the  weight  of  a 
cu.  ft.  of  lead  and  a  cu.  ft.  of  silver  ? 

28.  A  woman  who  had  learned  "  A  pint  is  a  pound,"  gave  a 
pint  of  shot  for  a  lb. ;  how  much  did  she  lose,  if  shot  is  15^  a  lb.  ? 

29.  How  many  gallons  of  air  will  weigh  one  pound  ? 


MISCELLANEOUS. 


155 


§  61.    Zero  and  Infinity. 


Numbers  may  be  regarded  as  existing 
in  three  realms. 

1.  Where  their  values  can  be  ex- 
pressed by  the  decimal  notation,  the 
finite. 

2.  Where  their  values  are  too  great  to 
be  expressed  by  the  decimal  notation, 
the  infinite. 

3.  Where  their  values  are  too  small 
to  be  expressed  by  the  decimal  notation, 
the  infinitesimal. 

Every  number  in  the  greatest  realm 
is  expressed  by  the  character  oc. 

This  does  not  stand  for  a  single 
number,  but  for  any  one  of  the 
countless  numbers  in  this  realm. 

Every  number  in  the  smallest  realm 
is  expressed  by  the  character  0. 

This  does  not  stand  for  a  single 
fraction,  but  for  any  one  of  the 
countless  fractions  in  this  realm. 


Illustration. 

The  number  of  ft.  in  a 
.mi.  can  be  expressed  by  the 
decimal  notation. 

The  number  of  cu.  in.  in 
space  is  too  great  to  be  ex- 
pressed by  the  decimal  no- 
tation. 

The  difference  between 
2  and  1.999-..,  where  9  is 
repeated  without  limit,  is 
too  small  to  be  expressed 
by  the  decimal  notation. 

One  cc  (infinity)  may  be 
2,  3,  or  any  other  number 
greater  or  less  than  another 
oc  ;  or  twice  three  times,  or 
any  number  of  times  as 
great. 

One  0  (infinitesimal)  may 
be  twice,  three  times,  or  any 
number  of  times  as  great  as 
another  0. 


30.  What  is  the  value  of  -  ? 

0 

31.  What  is  the  value  of  «  ? 

cc 

32.  What  is  the  value  of  -  ? 

6 

33.  What  is  the  value  of  |? 

34.  What  is  the  value  of  5.  ? 

QC 


Ans.  Any  finite  no.  as  2,  1000. 
Ans.  Any  no.  as  2,  100,  oc. 
Ans.  0. 

Ans.  cc.  The  smaller  the  divisor 
the  greater  the  quotient. 

Ans.  0.  The  larger  the  divisor 
the  smaller  the  quotient. 


GENERAL   REVIEW   EXERCISES. 


1.  The  sum  of  eight  numbers  is  95;  the  sum  of  seven  of  them 
is  87 ;  what  is  the  eighth  number  ? 

2.  The  addends  are  6,  8,  3,  9,  7,  4,  5,  6,  8 ;  what  is  the  sum  ? 

3.  The  subtrahend  is  986;  the  minuend  1000;  what  is  the 
remainder  ? 

4.  The  minuend  is  36 ;  the  rem.  12 ;  what  is  the  subtrahend  ? 

5.  The  multiplier  is  12;   the  multiplicand  13;    what  is  the 
product  ? 

6.  The  multiplicand  is  11 ;   the  product  132 ;   what  is  the 
multiplier  ? 

7.  The  dividend  is  144;  the  divisor  18;  what  is  the  quotient? 

8.  The  dividend  is  119 ;  the  quotient  9 ;  the  remainder  11 ; 
what  is  the  divisor  ? 

9.  The  dividend  is  125 ;  the  divisor  16 ;  what  is  the  remainder  ? 

10.  The  divisor  is  9 ;  the  quotient  13 ;  the  remainder  1 ;  what 
is  the  dividend  ? 

11.  What   number   multiplied   by   13,  with  7   added   to   the 
product,  will  give  85  ? 

12.  By  what  number  must  11  be  multiplied  so  that  when  4  is 
taken  from  the  result  the  remainder  will  be  128  ? 

13.  What  is  the  result  when  13  is  taken  7  times  as  an  addend  ? 

14.  How  many  times  must  12  be  taken  as  an  addend  to  pro- 
duce 108  ? 

15.  Define  a  prime  number ;   numbers  'prime   to   each   other ; 
numbers  severally  prime. 

16.  Name  three  composite  numbers  prime  to  each  other  but 
not  severally  prime. 


GENERAL  REVIEW   EXERCISES.  157 

17.  Give  the  rule  for  the  divisibility  of  a  number  by  2 ;  by  3 ; 
by  4 ;  by  5 ;  by  8  j  by  9 ;  by  11 ;  in  general. 

18.  Name  20  factors  of  180180. 

19.  77  and  91  are  factors  of  360360;  is  their  product  a 
factor  ?     Why  ? 

20.  Why  is  231  exactly  contained  in  360360  ? 

21.  Multiply  5x6x8  by  7,  and  express  the  result  by  its 
factors. 

22.  Divide  27  x  18  x  9  by  3,  and  express  the  result  by  its 
factors. 

23.  How  many  times  is  6x8x4x3  contained  in  48  x  36  ? 

24.  How  many  times  is  17  x  6  contained  in  51  x  2  x  3  ? 

25.  By  an  illustration,  show  that  the  remainder  found  by 
dividing  a  number  by  9,  is  the  same  as  the  remainder  found  by 
dividing  the  sum  of  its  digits  by  9. 

26.  By  an  illustration,  show  that  the  remainder  found  by  sub- 
tracting the  sum  of  its  digits  from  a  number  is  divisible  by  3. 

27.  Show  that  a  number  is  equal  to  its  digit  in  unit's  place, 
plus  ten  times  its  digit  in  ten's  place,  plus  one  hundred  times  its 
digit  in  hundred's  place,  and  so  on. 

28.  State  three  principles  for  finding  the  G.  C.  D. 

29.  By  the  second  principle,  how  can  you  tell  that  4  must  be 
the  G.  C.  D.  of  64  and  68  ? 

30.  By  the  third  principle,  how  can  you  tell  that  1  is  the 
G.  C.  D.  of  625  and  1728  ? 

31.  State  three  principles  for  finding  the  least  common  multiple. 

32.  Find  the  L.  C.  M.  of  20  and  30 ;  24  and  36  ;  30  and  35. 

33.  Find  the  L.  C.  M.  of  3,  8,  12,  24,  48,  72.  Did  you  use  the 
third  principle  ? 

34.  Analyze  and  explain  the  meaning  of  -f  by  the  first  concep- 
tion ;  by  the  second. 

35.  Which  of  these  two  methods  was  first  used  to  indicate  that 
5  is  to  be  divided  by  8,  5  -r-  8,  or  |  ? 


158  AMERICAN  MENTAL  ARITHMETIC. 

36.  14  -*-  3  =  4|.  Which  is  the  more  natural  conception,  that 
14  divided  by  3  equals  4  units  and  f  of  a  unit,  or  that  14  divided 
by  3  equals  4,  with  2  which  is  yet  to  be  divided  by  3  ? 

37.  Change  6}  to  an  improper  fraction ;  why  is  this  an  exam- 
ple in  addition  of  fractions  ? 

38.  Divide  17£  by  2\.     See  p.  63. 

39.  Divide  11\  by  2\  by  inverting  the  divisor  and  proceeding 
as  in  multiplication.  Is  this  process  as  easy  for  mental  work  as 
dividing  the  numerators  ? 

40.  What  is  the  difference  between  f  of  24  and  the  number  of 
which  9  is  f  ? 

41.  What  is  -|  and  J-  of  ^  of  6^  ?  Did  you  find  the  sum  of  ^ 
and  i  of  ^  before  you  multiplied  by  6J  ? 

42.  Reduce  ^2-  to  a  mixed  decimal,  and  express  the  result  in 
two  ways. 

43.  State  the  numeration  table  for  decimals. 

44.  Express  6  -=- 100  in  three  ways  ;  what  are  they  ? 

l 

45.  Is  there  any  difference  among  -£-,  .00^,  and  \°l0  ? 

46.  Eeduce  100  to  %. 

47.  What  month  of  the  year  is  January  ?  Oct.  ?  Dec.  ?  Aug.  ? 

48.  How  many  days  in  the  first  six  months  of  a  common  year  ? 

49.  State  the  number  of  cu.  in.  in  a  gal. ;  cu.  in.  in  a  bu. ;  rela- 
tion between  cu.  ft.  and  bu. ;  relation  between  cu.  ft.  and  gal. 

50.  How  many  cu.  ft.  of  hay  make  a  ton  ?  State  the  relation 
between  cu.  ft.  corn  in  the  ear  and  bu.  shelled  corn. 

51.  How  many  drops  make  1  teaspoonful  ?  gr.  make  1  lb.  troy  ? 
gr.  make  1  lb.  apothecaries'  ?  gr.  make  1  lb.  avoirdupois  ? 

52.  How  many  lb.  make  1  bu.  oats  ?  1  bu.  corn  ?  1  bu.  pota- 
toes ?  1  bu.  wheat  ? 

53.  In  the  metric  system,  state  tabulated  facts  about  the  unit 
of  long  measure  ;  unit  of  land  measure ;  unit  of  weight ;  unit  of 
capacity ;  unit  of  wood  measure. 


GENERAL  REVIEW  EXERCISES.  159 

54.  How  many  1.  in  1  Ml.  ?     How  many  1.  make  1  qt.  ? 

55.  How  many  pints  in  3  HI.  ?     How  many  1.  make  1  cu.  m.  ? 

56.  How  many  Ha.  in  2  acres  ? 

57.  How  many  cords  of  wood  in  40  steres  ? 

58.  Which  is  the  cheaper,  to  buy  meat  at  10^  a  lb.  or  at  20^  a 
Kg.  ?    By  how  much  a  lb.  ? 

59.  What  is  the  sum  in  lb.  of  a  common  English  ton,  a  long 
English  ton,  and  a  metric  ton  ? 

60.  How  much  is  made  per  quart  by  buying  chestnuts  at 
$  1.60  a  bu.  and  selling  at  5^  a  half-pint  ? 

61.  How  much  is  gained  per  lb.  by  buying  salt  at  $  20  a  ton 
and  selling  at  \$  an  oz.  ? 

62.  What  is  gained  on  6  dozen  eggs  by  buying  3  for  2^  and 
selling  2  for  3^  ? 

63.  By  buying  apples  at  2  for  a  cent,  and  the  same  number  at 
3  for  a  cent,  and  selling  all  at  5  for  2^,  I  lost  2^ ;  how  many 
apples  did  I  buy  ? 

64.  A  man  45  years  has  a  daughter  5  years  old.  In  how 
many  years  will  she  be  \  as  old  as  he  ?  \  as  old  ?  \  as  old  ?  Of 
the  same  age  ? 

65.  One  hunter  shot  24  pigeons,  another  shot  0.  The  first 
shot  how  many  times  as  many  as  the  second  ? 

66.  If  8  men  will  eat  a  quantity  of  flour  in  15  days,  how  long 
will  it  last  if  4  men  join  them  ? 

67.  A  loaned  $10  and  B  $  15  for  the  same  time  and  rate; 
together  they  received  $2  interest;  what  was  the  share  of 
each? 

68.  Three  men  hired  a  pasture  for  $9;  A  put  in  4  horses, 
B  6,  and  C  8  ;  what  ought  each  to  pay  ? 

69.  ^  of  a  cargo  was  lost ;  A,  who  owned  \  of  the  whole,  lost 
%  100  ;  what  was  the  value  of  the  part  that  remained  ? 

70.  If  a  man  can  do  f  of  a  piece  of  work  in  15  days,  how  long 
will  it  take  him  to  do  i  of  it  ? 


160  AMERICAN  MENTAL   ARITHMETIC. 

71.  If  a  hen  and  a  half  lay  an  egg  and  a  half  in  a  day  and  2 
half,  how  many  eggs  will  4  hens  lay  in  3  days  ? 

72.  If  3  cats  catch  3  rats  in  3  minutes,  how  many  cats  will  b( 
required  to  catch  100  rats  in  100  minutes  ? 

73.  A  thief  bought  a  pair  of  boots  for  $  5,  and  gave  in  pay 
ment  a  $  50  counterfeit  bill.  The  merchant  having'  no  money  a1 
all,  changed  the  bill  at  a  bank  and  gave  the  thief  $  45  in  gooc 
money.  After  the  merchant  had  paid  $  50  in  good  money  for  th( 
bill,  what  was  his  entire  loss  ? 

74.  The  freezing  and  boiling  points  in  the  Centigrade  ther 
mometer  are  0°  and  100°;  in  Fahrenheit's,  32°  and  212°.  Ho^ 
many  degrees  C.  equal  45°  F.  ? 

75.  How  many  degrees  F.  equal  45°  C.  ? 

76.  When  F.  reads  59°,  what  is  the  reading  of  C.  ? 

77.  When  F.  reads  23°,  what  is  the  reading  of  C.  ? 

78.  When  C.  reads  25°,  what  is  the  reading, of  F.  ? 

79.  When  C.  reads  10°  below  zero,  what  is  the  reading  of  F.  \ 

80.  By  selling  a  horse  for  $  36  a  man  gained  \  of  the  cost 
what  was  the  cost  ?     State  this  as  an  example  in  percentage. 

81.  By  selling  a  horse  for  $  60  a  man  lost  \  of  the  cost ;  whal 
was  the  cost  ?     State  this  as  an  example  in  percentage. 

82.  On  an  article  which  cost  $24  a  merchant  gained  33^%  : 
what  would  have  been  the  selling  price  if  he  had  gained  half  as 
much? 

83.  What  is  the  value  of  9.  ? 

oc 

84.  What  is  the  value  of  ^? 

85.  What  is  the  value  of  0  x  <x  ? 

86.  May  5  =  1?     May  5  =  2?     May  5  =  100000  ?     Why? 

87.  When  the,  diameter  of  a  circle  is  0,  what  is  its  circumfer- 
ence ?  What  is  the  ratio  of  its  circumference  to  its  diameter  ? 
What  is  the  value  of  this  ratio  ? 


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